Resolution of Singularities Steven Dale Cutkosky

Resolution of Singularities

Steven Dale Cutkosky

Department of Mathematics, University of Missouri, Columbia, Missouri 65211

To Hema, Ashok and Maya

Contents

Preface vii Chapter 1. Introduction 1 §1.1. Notation 2 Chapter 2. Non-singularity and Resolution of Singularities 3 §2.1. Newton’s method for determining the branches of a plane curve 3 §2.2. Smoothness and non-singularity 7 §2.3. Resolution of singularities 9 §2.4. Normalization 10 §2.5. Local uniformization and generalized resolution problems 11 Chapter 3. Curve Singularities 17 2 §3.1. Blowing up a point on A 17 §3.2. Completion 22 §3.3. Blowing up a point on a non-singular surface 25 §3.4. Resolution of curves embedded in a non-singular surface I 26 §3.5. Resolution of curves embedded in a non-singular surface II 29 Chapter 4. Resolution Type Theorems 37 §4.1. Blow-ups of ideals 37 §4.2. Resolution type theorems and corollaries 40 Chapter 5. Surface Singularities 45 §5.1. Resolution of surface singularities 45

v vi Contents

§5.2. Embedded resolution of singularities 56 Chapter 6. Resolution of Singularities in Characteristic Zero 61 §6.1. The operator 4 and other preliminaries 62 §6.2. Hypersurfaces of maximal contact and induction in resolution 66 §6.3. Pairs and basic objects 70 §6.4. Basic objects and hypersurfaces of maximal contact 75 §6.5. General basic objects 81 §6.6. Functions on a general basic object 83 §6.7. Resolution theorems for a general basic object 89 §6.8. Resolution of singularities in characteristic zero 99 Chapter 7. Resolution of Surfaces in Positive Characteristic 105 §7.1. Resolution and some invariants 105 §7.2. τ(q) = 2 109 §7.3. τ(q) = 1 113 §7.4. Remarks and further discussion 130 Chapter 8. Local Uniformization and Resolution of Surfaces 133 §8.1. Classification of valuations in function fields of dimension 2 133 §8.2. Local uniformization of algebraic function fields of surfaces 137 §8.3. Resolving systems and the Zariski-Riemann manifold 148 Chapter 9. Ramification of Valuations and Simultaneous Resolution 155 Appendix. Smoothness and Non-singularity II 163 §A.1. Proofs of the basic theorems 163 §A.2. Non-singularity and uniformizing parameters 169 §A.3. Higher derivations 171

§A.4. Upper semi-continuity of νq(I) 174 Bibliography 179 Index 185 Preface

The notion of singularity is basic to mathematics. In elementary algebra singularity appears as a multiple root of a polynomial. In geometry a point in a space is non-singular if it has a tangent space whose dimension is the same as that of the space. Both notions of singularity can be detected through the vanishing of derivitives. Over an algebraically closed field, a variety is non-singular at a point if there exists a tangent space at the point which has the same dimension as the variety. More generally, a variety is non-singular at a point if its local ring is a regular local ring. A fundamental problem is to remove a singularity by simple algebraic mappings. That is, can a given variety be desingularized by a proper, birational morphism from a non-singular variety? This is always possible in all dimensions, over fields of characteristic zero. We give a complete proof of this in Chapter 6. We also treat positive characteristic, developing the basic tools needed for this study, and giving a proof of resolution of surface singularities in positive characteristic in Chapter 7. In Section 2.5 we discuss important open problems, such as resolution of singularities in positive characteristic and local monomialization of morphisms. Chapter 8 gives a classification of valuations in algebraic function fields of surfaces, and a modernization of Zariski’s original proof of local uniformization for surfaces in characteristic zero. This book has evolved out of lectures given at the University of Mis- souri and at the Chennai Mathematics Institute, in Chennai, (also known as Madras), India. It can be used as part of a one year introductory sequence

vii viii Preface in algebraic geometry, and would provide an exciting direction after the basic notions of schemes and sheaves have been covered. A core course on resolution is covered in Chapters 2 through 6. The major ideas of resolution have been introduced by the end of Section 6.2, and after reading this far, a student will find the resolution theorems of Section 6.8 quite believable, and have a good feel for what goes into their proofs. Chapters 7 and 8 cover additional topics. These two chapters are independent, and can be chosen as possible followups to the basic material in the first 5 chapters. Chapter 7 gives a proof of resolution of singularities for surfaces in positive characteristic, and Chapter 8 gives a proof of local uniformization and resolution of singularities for algebraic surfaces. This chapter provides an introduction to valuation theory in algebraic geometry, and to the problem of local uniformization. The appendix proves foundational results on the singular locus that we need. On a first reading, I recommend that the reader simply look up the statements as needed in reading the main body of the book. Versions of almost all of these statements are much easier over algebraically closed fields of characteristic zero, and most of the results can be found in this case in standard textbooks in algebraic geometry. I assume that the reader has some familiarity with algebraic geometry and commutative algebra, such as can be obtained from an introductory course on these subjects. This material is covered in books such as Atiyah and MacDonald [13] or the basic sections of Eisenbud’s book [37], and the first two chapters of Hartshorne’s book on algebraic geometry [47], or Eisenbud and Harris’s book on schemes [38]. I thank Professors Seshadri and Ed Dunne for their encouragement to write this book, and Laura Ghezzi, TàiHà,Krishna Hanamanthu, Olga Kashcheyeva and Emanoil Theodorescu for their helpful comments on pre- liminary versions of the manuscript. For financial support during the preparation of this book I thank the National Science Foundation, the National Board of Higher Mathematics of India, the Mathematical Sciences Research Insititute and the University of Missouri.

Steven Dale Cutkosky Chapter 1

Introduction

An algebraic variety X is deﬁned locally by the vanishing of a system of polynomial equations fi ∈ K[x1, . . . , xn],

f1 = ··· = fm = 0.

If K is algebraically closed, points of X in this chart are α = (α1, . . . , αn) ∈ n AK which satisfy this system. The tangent space Tα(X) at a point α ∈ X n is the linear subspace of AK deﬁned by the system of linear equations

L1 = ··· = Lm = 0, where Li is deﬁned by n X ∂fi L = (α)(x − α ). i ∂x j j j=1 j

We have that dim Tα(X) ≥ dim X, and X is non-singular at the point α if dim Tα(X) = dim X. The locus of points in X which are singular is a proper closed subset of X. The fundamental problem of resolution of singularities is to perform simple algebraic transformations of X so that the transform Y of X is non- singular everywhere. To be precise, we seek a resolution of singularities of X; that is, a proper birational morphism Φ : Y → X such that Y is non-singular. The problem of resolution when K has characteristic zero has been studied for some time. In fact we will see (Chapters 2 and 3) that the method of Newton for determining the analytical branches of a plane curve singularity extends to give a proof of resolution for algebraic curves. The ﬁrst algebraic proof of resolution of surface singularities is due to Zariski [86]. We give

1 2 1. Introduction a modern treatment of this proof in Chapter 8. A study of this proof is surely the best introduction to the methods and ideas underlying the recent emphasis on valuation-theoretic methods in resolution problems. The existence of a resolution of singularities has been completely solved by Hironaka [52], in all dimensions, when K has characteristic zero. We give a simpliﬁed proof of this theorem, based on the proof of canonical resolution by Encinas and Villamayor ([40], [41]), in Chapter 6. When K has positive characteristic, resolution is known for curves, surfaces and 3-folds (with char(K) > 5). The ﬁrst proof in positive characteristic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar [1], [4]. We give several proofs, in Chapters 2, 3 and 4, of resolution of curves in arbitrary characteristic. In Chapter 5 we give a proof of resolution of surfaces in arbitrary characteristic.

1.1. Notation

The notation of Hartshorne [47] will be followed, with the following differ- ences and additions. By a variety over a field K (or a K-variety), we will mean an open subset n of an equidimensional reduced subscheme of the projective space PK . Thus an integral variety is a “quasi-projective variety” in the classical sense. A curve is a one-dimensional variety. A surface is a two-dimensional variety, and a 3-fold is a three-dimensional variety. A subvariety Y of a variety X is a closed subscheme of X which is a variety. An affine ring is a reduced ring which is of finite type over a field K. If X is a variety, and I is an ideal sheaf on X, we denote V (I) = spec(OX /I) ⊂ X. If Y is a subscheme of a variety X, we denote the ideal of Y in X by IY . If W1,W2 are subschemes of a variety of X, we will denote the scheme-theoretic intersection of W1 and W2 by W1 · W2. This is the subscheme

W1 · W2 = V (IW1 + IW2 ) ⊂ W. A hypersurface is a codimension one subvariety of a non-singular variety. Chapter 2

Non-singularity and Resolution of Singularities

2.1. Newton’s method for determining the branches of a plane curve

Newton’s algorithm for solving f(x, y) = 0 by a fractional power series 1 y = y(x m ) can be thought of as a generalization of the implicit function theorem to general analytic functions. We begin with this algorithm because of its simplicity and elegance, and because this method contains some of the most important ideas in resolution. We will see (in Section 2.5) that the algorithm immediately gives a local solution to resolution of analytic plane curve singularities, and that it can be interpreted to give a global solution to resolution of plane curve singularities (in Section 3.5). All of the proofs of resolution in this book can be viewed as generalizations of Newton’s algorithm, with the exception of the proof that curve singularities can be resolved by normalization (Theorems 2.14 and 4.3). Suppose K is an algebraically closed ﬁeld of characteristic 0, K[[x, y]] is a ring of power series in two variables and f ∈ K[[x, y]] is a non-unit, such P i j that x - f. Write f = i,j aijx y with aij ∈ K. Let

mult(f) = min{i + j | aij 6= 0}, and

mult(f(0, y)) = min{j | a0j 6= 0}.

3 4 2. Non-singularity and Resolution of Singularities

Set r0 = mult(f(0, y)) ≥ mult(f). Set i δ0 = min : j < r0 and aij 6= 0 . r0 − j r δ0 = ∞ if and only if f = uy 0 , where u is a unit in K[[x, y]]. Suppose that δ0 < ∞. Then we can write X i j f = aijx y

i+δ0j≥δ0r0 with a0r0 6= 0, and the weighted leading form

X i j r0 Lδ0 (x, y) = aijx y = a0r0 y + terms of lower degree in y i+δ0j=δ0r0 has at least two non-zero terms. We can thus choose 0 6= c1 ∈ K so that X j Lδ0 (1, c1) = aijc1 = 0. i+δ0j=δ0r0 Write δ = p0 , where q , p are relatively prime positive integers. We make 0 q0 0 0 a transformation q0 p0 x = x1 , y = x1 (y1 + c1). Then r0p0 f = x1 f1(x1, y1), where X j (2.1) f1(x1, y1) = aij(c1 + y1) + x1H(x1, y1).

i+δ0j=δ0r0

By our choice of c1, f1(0, 0) = 0. Set r1 = mult(f1(0, y1)). We see that r1 ≤ r0. We have an expansion X i j f1 = aij(1)x1y1. Set i δ1 = min : j < r1 and aij(1) 6= 0 , r1 − j and write δ = p1 with p , q relatively prime. We can then choose c ∈ K 1 q1 1 1 2 for f1, in the same way that we chose c1 for f, and iterate this process, obtaining a sequence of transformations

q0 p0 x = x1 , y = x1 (y1 + c1), q1 p1 (2.2) x1 = x2 , y1 = x2 (y2 + c2), . . Either this sequence of transformations terminates after a ﬁnite number n of steps with δn = ∞, or we can construct an inﬁnite sequence of transformations with δn < ∞ for all n. This allows us to write y as a series in ascending fractional powers of x. 2.1. Newton’s method for determining the branches of a plane curve 5

As our ﬁrst approximation, we can use our ﬁrst transformation to solve for y in terms of x and y1:

δ0 δ0 y = c1x + y1x . Now the second transformation gives us

δ1 δ1 δ0 δ0+ q δ0+ q y = c1x + c2x 0 + y2x 0 . We can iterate this procedure to get the formal fractional series

δ1 δ1 δ2 δ0 δ0+ q δ0+ q + q q (2.3) y = c1x + c2x 0 + c3x 0 0 1 + ··· .

Theorem 2.1. There exists an i0 such that δi ∈ N for i ≥ i0.

Proof. ri = mult(fi(0, yi)) are monotonically decreasing, and positive for all i, so it suﬃces to show that ri = ri+1 implies δi ∈ N. Without loss of generality, we may assume that i = 0 and r0 = r1. f1(x1, y1) is given by the expression (2.1). Set

X j g(t) = f1(0, t) = aij(c1 + t) .

i+δ0j=δ0r0 g(t) has degree r0. Since r1 = r0, we also have mult(g(t)) = r0. Thus r0 g(t) = a0r0 t , and

X j r0 aijt = a0r0 (t − c1) . i+δ0j=δ0r0 In particular, since K has characteristic 0, the binomial theorem shows that

(2.4) ai,r0−1 6= 0, where i is a natural number with i + δ0(r0 − 1) = δ0r0. Thus δ0 ∈ N. We can thus ﬁnd a natural number m, which we can take to be the smallest possible, and a series X i p(t) = bit such that (2.3) becomes

1 (2.5) y = p(x m ).

For n ∈ N, set n X i pn(t) = bit . i=1 Using induction, we can show that m mult(f(t , pn(t)) → ∞ 6 2. Non-singularity and Resolution of Singularities as n → ∞, and thus f(tm, p(t)) = 0. Thus X i (2.6) y = bix m is a branch of the curve f = 0. This expansion is called a Puiseux series (when r0 = mult(f)), in honor of Puiseux, who introduced this theory into algebraic geometry. Remark 2.2. Our proof of Theorem 2.1 is not valid in positive characteristic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over ﬁelds of positive characteristic. See Exercise 2.4 at the end of this section.

Suppose that f ∈ K[[x, y]] is irreducible, and that we have found a 1 solution y = p(x m ) to f(x, y) = 0. We may suppose that m is the smallest 1 natural number for which it is possible to write such a series. y − p(x m ) 1 divides f in R1 = K[[x m , y]]. Let ω be a primitive m-th root of unity in K. Since f is invariant under the K-algebra automorphism φ of R1 determined 1 1 j 1 by x m → ωx m and y → y, it follows that y − p(ω x m )) | f in R1 for all j, j 1 and thus y = p(ω x m ) is a solution to f(x, y) = 0 for all j. These solutions are distinct for 0 ≤ j ≤ m − 1, by our choice of m. The series m−1 Y j 1 g = y − p(ω x m ) j=0 is invariant under φ, so g ∈ K[[x, y]] and g | f in K[[x, y]], the ring of invariants of R1 under the action of the group Zm generated by φ. Since f is irreducible, m−1  Y j 1 f = u  y − p(ω x m )  , j=0 where u is a unit in K[[x, y]]. Remark 2.3. Some letters of Newton developing this idea are translated (from Latin) in [18].

After we have deﬁned non-singularity, we will return to this algorithm in (2.7) of Section 2.5, to see that we have actually constructed a resolution of singularities of a plane curve singularity. Exercise 2.4. 1. Construct a Puiseux series solution to f(x, y) = y4 − 2x3y2 − 4x5y + x6 − x7 = 0 over the complex numbers. 2.2. Smoothness and non-singularity 7

2. Apply the algorithm of this section to the equation f(x, y) = yp + yp+1 + x = 0 over an algebraically closed ﬁeld k of characteristic p. What is the resulting fractional series?

2.2. Smoothness and non-singularity

Deﬁnition 2.5. Suppose that X is a scheme. X is non-singular at P ∈ X if OX,P is a regular local ring.

Recall that a local ring R, with maximal ideal m, is regular if the dimension of m/m2 as an R/m vector space is equal to the Krull dimension of R. For varieties over a ﬁeld, there is a related notion of smoothness.

Suppose that K[x1, . . . , xn] is a polynomial ring over a ﬁeld K,

f1, . . . , fm ∈ K[x1, . . . , xn]. We deﬁne the Jacobian matrix  ∂f1 ··· ∂f1  ∂x1 ∂xn  .  J(f; x) = J(f1, . . . , fm; x1, . . . , xn) =  .  . ∂fm ··· ∂fm ∂x1 ∂xn

Let I = (f1, . . . , fm) be the ideal generated by f1, . . . , fm, and define R = K[x1, . . . , xn]/I. Suppose that P ∈ spec(R) has ideal m in K[x1, . . . , xn] and ideal n in R. Let K(P ) = Rn/nn. We will say that J(f; x) has rank l at P if the image of the l-th Fitting ideal Il(J(f; x)) of l × l minors of J(f, x) in K(P ) is K(P ) and the image of the s-th Fitting ideal Is(J(f; x)) in K(P ) is (0) for s > l. Definition 2.6. Suppose that X is a variety of dimension s over a field K, and P ∈ X. Suppose that U = spec(R) is an affine neighborhood of P such ∼ that R = K[x1, . . . , xn]/I with I = (f1, . . . , fm). Then X is smooth over K if J(f; x) has rank n − s at P .

This definition depends only on P and X and not on any of the choices of U, x or f. We verify this is the appendix. Theorem 2.7. Let K be a field. The set of points in a K-variety X which are smooth over K is an open set of X. Theorem 2.8. Suppose that X is a variety over a field K and P ∈ X. 1. Suppose that X is smooth over K at P . Then P is a non-singular point of X. 8 2. Non-singularity and Resolution of Singularities

2. Suppose that P is a non-singular point of X and K(P ) is separably generated over K. Then X is smooth over K at P .

In the case when K is algebraically closed, Theorems 2.7 and 2.8 are proven in Theorems I.5.3 and I.5.1 of [47]. Recall that an algebraically closed field is perfect. We will give the proofs of Theorems 2.7 and 2.8 for general fields in the appendix. Corollary 2.9. Suppose that X is a variety over a perfect field K and P ∈ X. Then X is non-singular at P if and only if X is smooth at P over K.

Proof. This is immediate since an algebraic function field over a perfect field K is always separably generated over K (Theorem 13, Section 13, Chapter II, [92]). In the case when X is an affine variety over an algebraically closed field K, the notion of smoothness is geometrically intuitive. Suppose that n X = V (I) = V (f1, . . . , fm) ⊂ AK is an s-dimensional affine variety, where I = (f1, . . . , fm) is a reduced and equidimensional ideal. We interpret the n closed points of X as the set of solutions to f1 = ··· = fm = 0 in K . We n identify a closed point p = (a1, . . . , an) ∈ V (I) ⊂ AK with the maximal ideal m = (x1 − a1, . . . , xn − an) of K[x1, . . . , xn]. For 1 ≤ i ≤ m, 2 fi ≡ fi(p) + Li mod m , where n X ∂fi L = (p)(x − a ). i ∂x j j j=1 i

But fi(p) = 0 for all i since p is a point of X. The tangent space to X in n AK at the point p is n Tp(X) = V (L1,...,Lm) ⊂ AK .

We see that dim Tp(X) = n − rank(J(f; x)(p)). Thus dim Tp(X) ≥ s (by Remark A.9), and X is non-singular at p if and only if dim Tp(X) = s. Theorem 2.10. Suppose that X is a variety over a ﬁeld K. Then the set of non-singular points of X is an open dense set of X.

This theorem is proven when K is algebraically closed in Corollary I.5.3 [47]. We will prove Theorem 2.10 when K is perfect in the appendix. The general case is proven in the corollary to Theorem 11 [85].

2 3 2 Exercise 2.11. Consider the curve y −x = 0 in AK , over an algebraically closed ﬁeld K of characteristic 0 or p > 3. 2.3. Resolution of singularities 9

1. Show that at the point p1 = (1, 1), the tangent space Tp1 (X) is the line −3(x − 1) + 2(y − 1) = 0.

2. Show that at the point p2 = (0, 0), the tangent space Tp2 (X) is the 2 entire plane AK . 3. Show that the curve is singular only at the origin p2.

2.3. Resolution of singularities

Suppose that X is a K-variety, where K is a ﬁeld. Deﬁnition 2.12. A resolution of singularities of X is a proper birational morphism φ : Y → X such that Y is a non-singular variety.

A birational morphism is a morphism φ : Y → X of varieties such that there is a dense open subset U of X such that φ−1(U) → U is an isomorphism. If X and Y are integral and K(X) and K(Y ) are the respective function fields of X and Y , φ is birational if and only if φ∗ : K(X) → K(Y ) is an isomorphism. A morphism of varieties φ : Y → X is proper if for every valuation ring V with morphism α : spec(V ) → X, there is a unique morphism β : spec(V ) → Y such that φ ◦ β = α. If X and Y are integral, and K(X) is the function field of X, then we only need consider valuation rings V such that K ⊂ V ⊂ K(X) in the definition of properness. The geometric idea of properness is that every mapping of a “formal” curve germ into X lifts uniquely to a morphism to Y . One consequence of properness is that every proper map is surjective. The properness assumption rules out the possibility of “resolving” by taking the birational resolution map to be the inclusion of the open set of non- singular points into the given variety, or the mapping of the non-singular points of a partial resolution to the variety. Birational proper morphisms of non-singular varieties (over a field of characteristic zero) can be factored by alternating sequences of blow-ups and blow-downs of non-singular subvarieties, as shown by Abramovich, Karu, Matsuki and Wlodarczyk [11]. We can extend our definition of a resolution of singularities to arbitrary schemes. A reasonable category to consider is excellent (or quasi-excellent) schemes (defined in IV.7.8 [45] and on page 260 of [66]). The definition of excellence is extremely technical, but the idea is to give minimal conditions ensuring that the the singular locus is preserved by natural base extensions such as completion. There are examples of non-excellent schemes which admit a resolution of singularities. Rotthaus [74] gives an example of a 10 2. Non-singularity and Resolution of Singularities regular local ring R of dimension 3 containing a field which is not excellent. In this case, spec(R) is a resolution of singularities of spec(R).

2.4. Normalization

Suppose that R is an affine domain with quotient field L. Let S be the integral closure of R in L. Since R is affine, S is finite over R, so that S is affine (cf. Theorem 9, Chapter V [92]). We say that spec(S) is the normalization of spec(R). Suppose that V is a valuation ring of L such that R ⊂ V . V is integrally closed in L (although V need not be Noetherian). Thus S ⊂ V . The morphism spec(V ) → spec(R) lifts to a morphism spec(V ) → spec(S), so that spec(S) → spec(R) is proper.

If X is an integral variety, we can cover X by open aﬃnes spec(Ri) with normalization spec(Si). The spec(Si) patch to a variety Y called the normalization of X. Y → X is proper, by our local proof. For a general (reduced but not necessarily integral) variety X, the normalization of X is the disjoint union of the normalizations of the irreducible components of X.

A ring R is normal if Rp is an integrally closed domain for all p ∈ spec(R). If R is an aﬃne ring and spec(S) is the normalization of spec(R) we constructed above, then S is a normal ring. Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if A is R1 (Ap is regular if ht(p) ≤ 1) and S2 (depth Ap ≥ min(ht(p), 2) for all p ∈ spec(A)).

A proof of this theorem is given in Theorem 23.8 [66]. Theorem 2.14. Suppose that X is a 1-dimensional variety over a ﬁeld K. Then the normalization of X is a resolution of singularities.

Proof. Let X be the normalization of X. All local rings OX,p of points p ∈ X are local rings of dimension 1 which are integrally closed, so Theorem 2.13 implies they are regular.

2 3 2 As an example, consider the curve singularity y − x = 0 in A , with affine ring R = K[x, y]/(y2 − x3). In the quotient field L of R we have the y 2 y y y relation ( x ) − x = 0. Thus x is integral over R. Since R[ x ] = K[ x ] is a y regular ring, it must be the integral closure of R in L. Thus spec(R[ x ]) → spec(R) is a resolution of singularities. Normalization is in general not enough to resolve singularities in dimension larger than 1. As an example, consider the surface singularity X defined 2.5. Local uniformization and generalized resolution problems 11

2 3 by z − xy = 0 in AK , where K is a field.√ The singular locus of X is defined by the ideal J = (z2 − xy, y, x, 2z). J = (x, y, z), so the singular locus of 2 2 2 X is the origin in A . Thus R = K[x, y, z]/(z − xy) is R1. Since z − xy is a regular element in K[x, y, z], R is Cohen-Macaulay, and hence R is S2 (Theorem 17.3 [66]). We conclude that X is normal, but singular. Kawasaki [58] has proven that under extremely mild assumptions a Noetherian scheme admits a Macaulayfication (all local rings have maximal depth). In general, Cohen-Macaulay schemes are far from being non- singular, but they do share many good homological properties with non- singular schemes. A scheme which does not admit a resolution of singularities is given by the example of Nagata (Example 3 in the appendix to [68]) of a one- dimensional Noetherian domain R whose integral closure R is not a finite R- module. We will show that such a ring cannot have a resolution of singularities. Suppose that there exists a resolution of singularities φ : Y → spec(R). That is, Y is a regular scheme and φ is proper and birational. Let L be the quotient field of R. Y is a normal scheme by Theorem 2.13. Thus Y ∼ has an open cover by affine open sets U1,...,Us such that Ui = spec(Ti), with Ti = R[gi1, . . . , git] where gij ∈ L, and each Ti is integrally closed. The integrally closed subring s \ T = Γ(Y, OY ) = Ti i=1 of L is a finite R-module since φ is proper (Theorem III, 3.2.1 [44] or Corol- lary II.5.20 [47] if φ is assumed to be projective). Exercise 2.15.

1. Let K = Zp(t), where p is a prime and t is an indeterminate. Let R = K[x, y]/(xp + yp − t), X = spec(R). Prove that X is non- singular, but there are no points of X which are smooth over K.

2. Let K = Zp(t), where p > 2 is a prime and t is an indeterminate. Let R = K[x, y]/(x2 + yp − t), X = spec(R). Prove that X is non-singular, and X is smooth over K at every point except at the prime (yp − t, x).

2.5. Local uniformization and generalized resolution problems

Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) of Section 2.1 determines an inclusion (2.7) R = K[[x, y]]/(f(x, y)) → K[[t]] 12 2. Non-singularity and Resolution of Singularities with x = tm, y = p(t) the series of (2.5). Since the m chosen in (2.5) of Section 2.1 is the smallest possible, we can verify that t is in the quotient field L of R. Since the quotient field of K[[t]] is generated by 1, t, . . . , tm−1 over L, we conclude that R and K[[t]] have the same quotient field. (2.7) is an explicit realization of the normalization of the curve singularity germ f(x, y) = 0, and thus spec(K[[t]]) → spec(R) is a resolution of singularities. The quotient field L of K[[t]] has a valuation ν defined for non-zero h ∈ L by

ν(h(t)) = n ∈ Z if h(t) = tnu, where u is a unit in K[[t]]. This valuation can be understood in R by the Puiseux series (2.6). More generally, we can consider an algebraic function field L over a field K. Suppose that V is a valuation ring of L containing K. The problem of local uniformization is to find a regular local ring R, essentially of finite type over K and with quotient field L such that the valuation ring V dominates R (R ⊂ V and the intersection of the maximal ideal of V with R is the maximal ideal of R). The exact relationship of local uniformization to resolution of singularities is explained in Section 8.3. If L has dimension 1 over K, then the valuation rings V of L which contain K are precisely the local rings of points on the unique non-singular projective curve C with function field L (cf. Section I.6 [47]). The Newton method of Section 2.1 can be viewed as a solution to the local uniformization problem for complex analytic curves. If L has dimension ≥ 2 over K, there are many valuations rings V of L which are non-Noetherian (see the exercise of Section 8.1). Zariski proved local uniformization for two-dimensional function fields over an algebraically closed field of characteristic zero in [86]. He was able to patch together local solutions to prove the existence of a resolution of singularities for algebraic surfaces over an algebraically closed field of characteristic zero. He later was able to prove local uniformization for algebraic function fields of characteristic zero in [87]. This method leads to an extremely difficult patching problem in higher dimensions, which Zariski was able to solve in dimension 3 in [90]. Zariski’s proof of local uniformization can be considered as an extension of the Newton method to general valuations. In Chapter 8, we present Zariski’s proof of resolution of surface singularities through local uniformization. Local uniformization has been proven for two-dimensional function fields in positive characteristic by Abhyankar. Abhyankar has proven resolution of singularities in positive characteristic for surfaces and for three-dimensional varieties, [1],[3],[4]. 2.5. Local uniformization and generalized resolution problems 13

There has recently been a resurgence of interest in local uniformization in positive characteristic, and significant progress is being made. Some of the recent papers on this area are: Hauser [50], Heinzer, Rotthaus and Wiegand [51], Kuhlmann [59], [60], Piltant [71], Spivakovsky [77], Teissier [78]. Some related resolution type problems are resolution of vector fields, resolution of differential forms and monomialization of morphisms. Suppose that X is a variety which is smooth over a perfect field K, and 1 D ∈ Hom(ΩX/K , OX ) is a vector field. If p ∈ X is a closed point, and x1, . . . , xn are regular parameters at p, then there is a local expression ∂ ∂ D = a1(x) + ··· + an(x) , ∂x1 ∂xn where ai(x) ∈ OX,p. We can associate an ideal sheaf ID to D on X by

ID,p = (a1, . . . , an) for p ∈ X.

There are an effective divisor FD and an ideal sheaf JD such that V (JD) has codimension ≥ 2 in X and ID = OX (−FD)JD. The goal of resolution of vector fields is to find a proper morphism π : Y → X so that if D0 = π−1(D), then the ideal ID0 ⊂ OY is as simple as possible. This was accomplished by Seidenberg for vector fields on non-singular surfaces over an algebraically closed field of characteristic zero in [75]. The best statement that can be attained is that the order of JD0 at p (Definition A.17) is νp(JD0 ) ≤ 1 for all p ∈ Y . The basic invariant considered in the proof is the order νp(JD). While this is an upper semi-continuous function on Y , it can go up under a monoidal transform. However, we have that −1 νq(JD0 ) ≤ νp(JD)+1 if π : Y → X is the blow-up of p, and q ∈ π (p). This should be compared with the classical resolution problem, where a basic result is that order cannot go up under a permissible monoidal transform (Lemma 6.4). Resolution of vector fields for smooth surfaces over a perfect field has been proven by Cano [19]. Cano has also proven a local theorem for resolution for vector fields over fields of characteristic zero [20], which implies local resolution (along a valuation) of a vector field. The statement is that we can achieve νq(JD0 ) ≤ 1 (at least locally along a valuation) after a morphism π : Y → X. There is an analogous problem for differential forms. A recent paper on this is [21]. We can also consider resolution problems for morphisms f : Y → X of varieties. The natural question to ask is if it is possible to perform monoidal transforms (blow-ups of non-singular subvarieties) over X and Y to produce a morphism which is a monomial mapping. 14 2. Non-singularity and Resolution of Singularities

Definition 2.16. Suppose that Φ : X → Y is a dominant morphism of non- singular irreducible K-varieties (where K is a field of characteristic zero). Φ is monomial if for all p ∈ X there exists an étaleneighborhood U of p, uniformizing parameters (x1, . . . , xn) on U, regular parameters (y1, . . . , ym) in OY,Φ(p), and a matrix (aij) of non-negative integers (which necessarily has rank m) such that

a11 a1n y1 = x1 ··· xn , . . am1 amn ym = x1 ··· xn . Definition 2.17. Suppose that Φ : X → Y is a dominant morphism of integral K-varieties. A morphism Ψ : X1 → Y1 is a monomialization of Φ [27] if there are sequences of blow-ups of non-singular subvarieties α : X1 → X and β : Y1 → Y , and a morphism Ψ : X1 → Y1 such that the diagram Ψ X1 → Y1 ↓ ↓ X →Φ Y commutes, and Ψ is a monomial morphism. If Φ : X → Y is a dominant morphism from a 3-dimensional variety to a surface (over an algebraically closed field of characteristic 0), then there is a monomialization of Φ [27]. A generalized multiplicity is defined in this paper, and it can go up, causing a very high complexity in the proof. An extension of this result to strongly prepared morphisms from n-folds to surfaces is proven in [33]. It is not known if monomialization is true even for birational morphisms of varieties of dimension ≥ 3, although it is true locally along a valuation, from the following Theorem 2.18. Theorem 2.18 is proven when the quotient field of S is finite over the quotient field of R in [25]. The proof for general field extensions is in [28]. Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R ⊂ S are regular local rings, essentially of finite type over a field K of characteristic zero. Let V be a valuation ring of K which dominates S. Then there exist sequences of monoidal transforms R → R0 and S → S0 such that V domi- 0 0 0 nates S , S dominates R and there are regular parameters (x1, ...., xm) in 0 0 0 R , (y1, ..., yn) in S , units δ1, . . . , δm ∈ S and an m × n matrix (aij) of non-negative integers such that rank(aij) = m is maximal and

a11 a1n x1 = y1 .....yn δ1, . (2.8) . am1 amn xm = y1 .....yn δm. 2.5. Local uniformization and generalized resolution problems 15

Thus (since char(K) = 0) there exists an etale extension S0 → S00, 00 where S has regular parameters y1,..., yn such that x1, . . . , xm are pure monomials in y1,..., yn. The standard theorems on resolution of singularities allow one to easily find R0 and S0 such that (2.8) holds, but, in general, we will not have the essential condition rank(aij) = m. The difficulty of the problem is to achieve this condition. This result gives very simple structure theorems for the ramification of valuations in characteristic zero function fields [35]. We discuss some of these results in Chapter 9. A generalization of monomialization in characteristic p function fields of algebraic surfaces is obtained in [34] and especially in [35]. We point out that while it seems possible that Theorem 2.18 does hold in positive characteristic, there are simple examples in positive characteristic where a monomialization does not exist. The simplest example is the map of curves

y = xp + xp+1 in characteristic p. A quasi-complete variety over a field K is an integral finite type K- scheme which satisfies the existence part of the valuative criterion for properness (Hironaka, Chapter 0, Section 6 of [52] and Chapter 8 of [26]). The construction of a monomialization by quasi-complete varieties follows from Theorem 2.18. Theorem 2.19 is proven for generically finite morphisms in [26] and for arbitrary morphisms in Theorem 1.2 [28].

Theorem 2.19 (Theorem 1.2 [26],Theorem 1.2 [28]). Let K be a ﬁeld of characteristic zero, Φ: X → Y a dominant morphism of proper K-varieties. Then there exist birational morphisms of non-singular quasi-complete K- varieties α : X1 → X and β : Y1 → Y , and a monomial morphism Ψ: X1 → Y1 such that the diagram

Ψ X1 → Y1 ↓ ↓ Φ X → Y commutes and α and β are locally products of blow-ups of non-singular subvarieties. That is, for every z ∈ X1, there exist affine neighborhoods V1 of z and V of x = α(z) such that α : V1 → V is a finite product of monoidal transforms, and there exist affine neighborhoods W1 of Ψ(z), W of y = β(Ψ(z)) such that β : W1 → W is a finite product of monoidal transforms. 16 2. Non-singularity and Resolution of Singularities

A monoidal transform of a non-singular K-scheme S is the map T → S induced by an open subset T of proj(L In), where I is the ideal sheaf of a non-singular subvariety of S. The proof of Theorem 2.19 follows from Theorem 2.18, by patching a ﬁnite number of local solutions. The resulting schemes may not be separated. It is an extremely interesting question to determine if a monomialization exists for all morphisms of varieties (over a ﬁeld of characteristic zero). That is, the conclusions of Theorem 2.19 hold, but with the stronger conditions that α and β are products of monoidal transforms on proper varieties X1 and Y1. Chapter 3

Curve Singularities

3.1. Blowing up a point on A2 Suppose that K is an algebraically closed field. Let 2 2 U1 = spec(K[s, t]) = AK ,U2 = spec(K[u, v]) = AK . Define a K-algebra isomorphism 1 1 λ : K[s, t] = K[s, t, ] → K[u, v] = K[u, v, ] s s v v 1 by λ(s) = v , λ(t) = uv. We define a K-variety B0 by patching U1 to U2 on the open sets

U2 − V (v) = spec(K[u, v]v) and U1 − V (s) = spec(K[s, t]s) by the isomorphism λ. The K-algebra homomorphisms K[x, y] → K[s, t] deﬁned by x 7→ st, y 7→ t, and K[x, y] → K[u, v] deﬁned by x 7→ u, y 7→ uv, are compatible with the isomorphism λ, so we get a morphism 2 π : B(p) = B0 → U0 = spec(K[x, y]) = AK , where p denotes the origin of U0. Upon localization of the above maps, we get isomorphisms ∼ ∼ K[x, y]y = K[s, t]t and K[x, y]x = K[u, v]u.

17 18 3. Curve Singularities

∼ ∼ Thus π : U1 − V (t) = U0 − V (y), π : U2 − V (u) = U0 − V (x), and π is an isomorphism over U0 − V (x, y) = U0 − {p}. Moreover, −1 π (p) ∩ U1 = V (st, t) = {V (t) ⊂ spec(K[s, t])} = spec(K[s]), 1 π−1(p) ∩ U = V (u, uv) = {V (u) ⊂ spec(K[u, v])} = spec(K[ ]), 2 s 1 −1 ∼ 1 −1 by the identiﬁcation s = v . Thus π (p) = P . Set E = π (p). We have 1 “blown up p” into a codimension 1 subvariety of B(p), isomorphic to P . Set R = K[x, y], m = (x, y). Then x x U = spec(K[s, t]) = spec(K[ , y]) = spec(R[ ]), 1 y y y y U = spec(K[x, ]) = spec(R[ ]). 2 x x We see that x y M B(p) = spec(R[ ]) ∪ spec(R[ ]) = proj( mn). y x n≥0

−1 Suppose that q ∈ π (p) is a closed point. If q ∈ U1, then its associated x ideal mq is a maximal ideal of R1 = K[ y , y] which contains (x, y)R1 = yR1. x x Thus mq = (y, y − α) for some α ∈ K. If we set y1 = y, x1 = y − α, we see that there are regular parameters (x1, y1) in OB(p),q such that

x = y1(x1 + α), y = y1. −1 By a similar calculation, if q ∈ π (p) and q ∈ U2, there are regular parameters (x1, y1) in OB(p),q such that

x = x1, y = x1(y1 + β) for some β ∈ K. If the constant α or β is non-zero, then q is in both U1 −1 and U2. Thus the points in π (p) can be expressed (uniquely) in one of the forms

x = x1, y = x1(y1 + α) with α ∈ K, or x = x1y1, y = y1. Since B(p) is projective over spec(R), it certainly is proper over spec(R). However, it is illuminating to give a direct proof. Lemma 3.1. B(p) → spec(R) is proper.

y x Proof. Suppose that V is a valuation ring containing R. Then x or y ∈ V . y y Say x ∈ V . Then R[ x ] ⊂ V , and we have a morphism y spec(V ) → spec(R[ ]) ⊂ B(p) x which lifts the morphism spec(V ) → spec(R). 2 3.1. Blowing up a point on A 19

More generally, suppose that S = spec(R) is an affine surface over a field L, and p ∈ S is a non-singular closed point. After possibly replacing S with an open subset spec(Rf ), we may assume that the maximal ideal of p in R is mp = (x, y). We can then define the blow up of p in S by M n (3.1) π : B(p) = proj( mp ) → S. n≥0 We can write B(p) as the union of two affine open subsets: x y B(p) = spec(R[ ]) ∪ spec(R[ ]). y x −1 ∼ 1 π is an isomorphism over S − p, and π (p) = P . Suppose that S is a surface, and p ∈ S is a non-singular point, with ideal sheaf mp ⊂ OS. The blow-up of p ∈ S is M n π : B(p) = proj( mp ) → S. n≥0 π is an isomorphism away from p, and if U = spec(R) ⊂ S is an affine open neighborhood of p in S such that

(x, y) = Γ(U, mp) ⊂ R is the maximal ideal of p in R, then the map π : π−1(U) → U is deﬁned by the construction (3.1). 2 Suppose that C ⊂ AK is a curve. Since K[x, y] is a unique factorization domain, there exists f ∈ K[x, y] such that V (f) = C. If q ∈ C is a closed point, we will denote the corresponding maximal ideal of K[x, y] by mq. We set r νq(C) = max{r | f ∈ mq}

(more generally, see Deﬁnition A.17). νq(C) is both the multiplicity and the order of C at q.

2 Lemma 3.2. q ∈ AK is a non-singular point of C if and only if νq(C) = 1.

Proof. Let R = K[x, y]mq , and suppose that mq = (x, y). Let T = R/fR. 2 2 ∼ 2 If νq(C) ≥ 2, then f ∈ mq, and mqT/mqT = mq/mq has dimension 2 > 1, so that T is not regular and q is singular on C. However, if νq(C) = 1, then we have 2 f ≡ αx + βy mod mq, where α, β ∈ K and at least one of α and β is non-zero. Without loss of generality, we may suppose that β 6= 0. Thus α y ≡ − x mod m2 + (f), β q 20 3. Curve Singularities

2 ∼ 2 and x is a K generator of mqT/mqT = mq/mq + (f). We see that 2 dimK mqT/mqT = 1 and q is non-singular on C. Remark 3.3. This lemma is also true in the situation where q is a non- singular point on a non-singular surface S (over a ﬁeld K) and C is a curve contained in S. We need only modify the proof by replacing R with the regular local ring R = OS,q, which has regular parameters (x, y) which are 2 a K(q) basis of mq/mq, and since R is a unique factorization domain, there is f ∈ R such that f = 0 is a local equation of C at q.

2 Let p be the origin in AK , and suppose that νp(C) = r > 0. Let 2 ˜ π : B(p) → AK be the blow-up of p. The strict transform C of C in B(p) is the Zariski closure of π−1(C − p) =∼ C − p in B(p). Let E = π−1(p) be the exceptional divisor . Set-theoretically, π−1(C) = C˜ ∪ E.

For some aij ∈ K, we have a ﬁnite sum X i j f = aijx y i+j≥r with aij 6= 0 for some i, j with i + j = r. In the open subset U2 = spec(K[x1, y1]) of B(p), where

x = x1, y = x1y1, x1 = 0 is a local equation for E. Also, r f = x1f1, where X i+j−r j f1 = aijx1 y1. i+j≥r x1 - f1 since νp(C) = r, so that f1 = 0 is a local equation of the strict transform C˜ of C in U2.

In the open subset U1 = spec(K[x1, y1]) of B(p), where

x = x1y1, y = y1, y1 = 0 is a local equation for E and r f = y1f 1, where X i i+j−r f 1 = aijx1y1 i+j≥r is a local equation of C˜ in U1. 2 3.1. Blowing up a point on A 21

As a scheme, we see that π−1(C) = C˜ + rE. The scheme-theoretic preimage of C is called the total transform of C. Deﬁne the leading form of f to be X i j L = aijx y . i+j=r −1 Suppose that q ∈ π (p). There are regular parameters (x1, y1) at q of one of the forms

x = x1, y = x1(y1 + α) or x = x1y1, y = y1. f ˜ In the ﬁrst case f1 = r = 0 is a local equation of C at q, where x1 X j f1 = aij(y1 + α) + x1Ω i+j=r for some polynomial Ω. Thus νq(C˜) ≤ r, and νq(C˜) = r implies X j r aij(y1 + α) = a0ry1. i+j=r We then see that X j r aijy1 = a0r(y1 − α) i+j=r and r L = a0r(y − αx) . f ˜ In the second case f1 = r = 0 is a local equation of C at q, where y1 X i f1 = aijx1 + y1Ω i+j=r for some series Ω. Thus νq(C˜) ≤ r, and νq(C˜) = r implies X i r aijx1 = ar0x1 i+j=r and r L = ar0x . −1 We then see that there exists a point q ∈ π (p) such that νq(C˜) = r only when L = (ax + by)r for some constants a, b ∈ K. Since r > 0, there is at most one point q ∈ π1(p) where the multiplicity does not drop.

Exercise 3.4. Suppose that K[x1, . . . , xn] is a polynomial ring over a perfect ﬁeld K and f ∈ K[x1, . . . , xn] is such that f(0,..., 0) = 0. Let m = (x1, . . . , xn), a maximal ideal of K[x1, . . . , xn].

Deﬁne ord f(0,..., 0, xn) = r if r r+1 xn | f(0,..., 0, xn) and xn - f(0,..., 0, xn). 22 3. Curve Singularities

Show that R = (K[x1, . . . , xn]/(f))m is a regular local ring if ord f(0,..., 0, xn) = 1.

3.2. Completion

Suppose that A is a local ring with maximal ideal m. A coefficient field of A is a subfield L of A which is mapped onto A/m by the quotient mapping A → A/m. A basic theorem of Cohen is that an equicharacteristic complete local ring contains a coefficient field (Theorem 27 Section 12, Chapter VIII [92]). This leads to Cohen’s structure theorem (Corollary, loc. cit.), which shows that an equicharacteristic complete regular local ring A is isomorphic to a formal power series ring over a field. In fact, if L is a coefficient field of A, and if (x1, . . . , xn) is a regular system of parameters of A, then A is the power series ring

A = L[[x1, . . . , xn]]. We further remark that the completion of a local ring R is a regular local ring if and only if R is regular (cf. Section 11, Chapter VIII [92]). Lemma 3.5. Suppose that S is a non-singular algebraic surface defined over a field K, p ∈ S is a closed point, π : B = B(p) → S is the blow-up of p, and suppose that q ∈ π−1(p) is a closed point such that K(q) is separable over K(p). Let R1 = OˆS,p and R2 = OˆB,q, and suppose that K1 is a coefficient field of R1, (x, y) are regular parameters in R1. Then there exist a coefficient field K2 = K1(α) of R2 and regular parameters (x1, y1) of R2 such that ∗ πˆ : R1 → R2 is the map given by

X i j X i+j j aijx y → aijx1 (y1 + α) , i,j≥0 i,j≥0 where aij ∈ K1, or K2 = K1 and ∗ πˆ : R1 → R2 is the map given by

X i j X i i+j aijx y → aijx1y1 . i,j≥0 i,j≥0

Proof. We have R1 = K1[[x, y]], and a natural homomorphism

OS → OS,p → R1 3.2. Completion 23 which induces y x q ∈ B(p) × spec(R ) = spec(R [ ]) ∪ spec(R [ ]). S 1 1 x 1 y y Let mq be the ideal of q in B(p) ×S spec(R1). If q ∈ spec(R1[ x ]), then y y y K(q) =∼ R [ ]/m =∼ K [ ]/(f( )) 1 x q 1 x x y y for some irreducible polynomial f( x ) in the polynomial ring K1[ x ]. Since ∼ y K(q) is separable over K1 = K(p), f is separable. We have mq = (x, f( x )) ⊂ y R1[ x ]. y y Let α ∈ R1[ x ]/mq be the class of x . We have the residue map φ : R2 → L y where L = R1[ x ]/mq. There is a natural embedding of K1 in R2. We have a factorization of f(t) = (t − α)γ(t) in L[t], where t − α and γ(t) are relatively prime. By Hensel’s Lemma (Theorem 17, Section 7, Chapter VIII [92]) there is α ∈ R2 such that φ(α) = α and f(t) = (t − α)γ(t) in R2[t], where φ(γ(t)) = γ(t). The subfield K2 = K1[α] of R2 is thus a coefficient field of R2, and mqR2 = y y y (x, x − α). Thus R2 = K2[[x, x − α]]. Set x1 = x, y1 = x − α. The inclusion

R1 = K1[[x, y]] → R2 = K2[[x1, y1]] is natural. A series X i j aijx y with coeﬃcients aij ∈ K1 maps to the series

X i+j j aijx1 (y1 + α) .

We now give an example to show that even if a regular local ring contains a field, there may not be a coefficient field of the completion of the ring containing that field. Thus the above lemma does not extend to non-perfect fields.

Let K = Zp(t), where t is an indeterminate. Let R = K[x](xp−t). Sup- pose that Rˆ has a coeﬃcient ﬁeld L containing K. Let φ : Rˆ → R/mˆ be the residue map. Since φ | L is an isomorphism, there exists λ ∈ L such that λp = t. Thus (xp − t) = (x − λ)p in Rˆ. But this is impossible, since (xp − t) is a generator of mRˆ, and is thus irreducible.

Theorem 3.6. Suppose that R is a reduced aﬃne ring over a ﬁeld K, and A = Rp, where p is a prime ideal of R. Then the completion Aˆ = Rˆp of A with respect to its maximal ideal is reduced. 24 3. Curve Singularities

When K is a perfect field, this is a theorem of Chevalley (Theorem 31, Section 13, Chapter VIII [92]). The general case follows from Scholie IV 7.8.3 (vii) [45]. However, the property of being a domain is not preserved under com- 2 2 3 pletion. A simple example is f = y − x + x . f is irreducible in C[x, y], but is reducible in the completion C[[x, y]]: √ √ f = y2 − x2 − x3 = (y − x 1 + x)(y + x 1 + x). The first parts of the expansions of the two factors are 1 y − x − x2 + ··· 2 and 1 y + x + x2 + ··· . 2 Lemma 3.7 (Weierstrass Preparation Theorem). Let K be a field, and suppose that f ∈ K[[x1, . . . , xn, y]] is such that 0 < r = ν(f(0,..., 0, y)) = max{n | yn divides f(0,..., 0, y)} < ∞.

Then there exist a unit series u in K[[x1, . . . , xn, y]] and non-unit series ai ∈ K[[x1, . . . , xn]] such that r r−1 f = u(y + a1y + ··· + ar). A proof is given in Theorem 5, Section 1, Chapter VII [92]. A concept which will be important in this book is the Tschirnhausen transformation, which generalizes the ancient notion of “completion of the square” in the solution of quadratic equations. Deﬁnition 3.8. Suppose that K is a ﬁeld of characteristic p ≥ 0 and f ∈ K[[x1, . . . , xn, y]] has an expression r r−1 f = y + a1y + ··· + ar with ai ∈ K[[x1, . . . , xn]] and p = 0 or p - r. The Tschirnhausen transformation of f is the change of variables replacing y with a y0 = y + 1 . r f then has an expression 0 r 0 r−2 f = (y ) + b2(y ) + ··· + br with bi ∈ K[[x1, . . . , xn]] for all i.

Exercise 3.9. Suppose that (R, m) is a complete local ring and L1 ⊂ R is a field such that R/m is finite and separable over L1. Use Hensel’s Lemma (Theorem 17, Section 7, Chapter VIII [92]) to prove that there exists a coefficient field L2 of R containing L1. 3.3. Blowing up a point on a non-singular surface 25

3.3. Blowing up a point on a non-singular surface

We define the strict transform, the total transform, and the multiplicity (or order) νp(C) analogously to the definition of Section 3.1 (More generally, see Section 4.1 and Definition A.17). Lemma 3.10. Suppose that X is a non-singular surface over an algebraically closed field K, and C is a curve on X. Suppose that p ∈ X and νp(C) = r. Let π : B(p) → X be the blow-up of p, C˜ the strict transform of −1 C, and suppose that q ∈ π (p). Then νq(C˜) ≤ r, and if r > 0, there is at −1 most one point q ∈ π (p) such that νq(C˜) = r. Proof. Let f = 0 be a local equation of C at p. If (x, y) are regular parameters in OX,p, we can write X i j f = aijx y ∼ as a series of order r in OˆX,p = K[[x, y]]. Let X i j L = aijx y i+j=r be the leading form of f. −1 If q ∈ π (p), then OB(p),q has regular parameters (x1, y1) such that

x = x1, y = x1(y1 + α), or x = x1y1, y = y1. This substitution induces the inclusion ˆ ˆ K[[x, y]] = OX,p → OB(p),q = K[[x1, y1]].

First suppose that (x1, y1) are deﬁned by

x = x1, y = x1(y1 + α). ˜ f Then a local equation for C at q is f1 = r , which has the expansion x1 X j f1 = aij(y1 + α) + x1Ω for some series Ω. We ﬁnish the proof as in Section 3.1. 2 3 2 Suppose that C is the curve y − x = 0 in AK . The only singular point 2 ˜ of C is the origin p. Let π : B(p) → AK be the blow-up of p, C the strict transform of C, and E the exceptional divisor E = π−1(p). x On U1 = spec(K[ y , y]) ⊂ B(p) we have coordinates x1, y1 with x = x1y1, y = y1. A local equation for E on U1 is y1 = 0. Also, 2 3 2 3 y − x = y1(1 − x1y1). ˜ 3 A local equation for C on U1 is 1 − x1y1 = 0, which is a unit on E ∩ U1, so C˜ ∩ E ∩ U1 = ∅. 26 3. Curve Singularities

y On U2 = spec(K[x, x ]) ⊂ B(p) we have coordinates x1, y1 with x = x1, y = x1y1. A local equation for E on U2 is x1 = 0. Also, 2 3 2 2 y − x = x1(y1 − x1). ˜ 2 A local equation for C on U2 is y1 −x1 = 0, which has order ≤ 1 everywhere. Thus C˜ is non-singular, and C˜ → C is a resolution of singularities. Finally, y Γ(C,˜ O ) = K[x , y ]/(y2 − x ) = K[y ] = K[ ]. C˜ 1 1 1 1 1 x This is the normalization of K[x, y]/(y2 − x3) that we computed in Section 2.4. As a further example, consider the curve C with equation y2 − x5 in 2 2 A . The only singular point of C is the origin p. Let π : B(p) → A be the blow-up of p, E be the exceptional divisor, C˜ be the strict transform of C. There is only one singular point q on C˜. Regular parameters at q are (x1, y1), where x = x1, y = x1y1. C˜ has the local equation 2 3 y1 − x1 = 0.

In this example νq(C˜) = νp(C) = 2, so the multiplicity has not dropped. However, this singularity is resolved after blowing up q, as we calculated in the previous example. These examples suggest an algorithm to resolve curve singularities. First blow up all singular points. If the resulting curve is not resolved, blow up the new singular points, and repeat as long as the curve is singular. This algorithm ends after a finite number of iterations in a non-singular curve. In Sections 3.4 and 3.5 we will prove this for curves embedded in a non-singular surface, first over an algebraically closed field K of characteristic zero, and then over an arbitrary field.

3.4. Resolution of curves embedded in a non-singular surface I

In this section we consider curves embedded in a non-singular surface over an algebraically closed ﬁeld K of characteristic zero, and prove that their singularities can be resolved. The theorem is proved by passing to the completion of the local ring of the surface at a singular point of the curve. This allows us to view a local equation of the curve as a power series in two variables. Our main invariant is the multiplicity of the curve at a point. We know that this multiplicity 3.4. Resolution of curves embedded in a non-singular surface I 27 cannot go up after blowing up, so we must show that it will eventually go down, after enough blowing up. Theorem 3.11. Suppose that C is a curve on a non-singular surface X over an algebraically closed ﬁeld K of characteristic zero. Then there exists a sequence of blow ups of points λ : Y → X such that the strict transform C˜ of C on Y is non-singular.

Proof. Let r = max(νp(C) | p ∈ C}. If r = 1, C is non-singular, so we may assume that r > 1. The set {p ∈ C | ν(p) = r} is a subset of the singular locus of C, which is a proper closed subset of the 1-dimensional variety C, so it is a ﬁnite set. The proof is by induction on r. We can construct a sequence of projective morphisms

πn π1 (3.2) · · · → Xn → · · · → X0 = X, where each πn+1 : Xn+1 → Xn is the blow-up of all points on the strict transform Cn of C which have multiplicity r on Cn. If this sequence is finite, then there is an integer n such that all points on the strict transform Cn of C have multiplicity ≤ r − 1. By induction on r, we can repeat this process to construct the desired morphism Y → X which induces a resolution of C. We will assume that the sequence (3.2) is infinite, and derive a contradiction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cn which have multiplicity r on Cn and such that πn+1(pn+1) = pn for all n. ˆ Let Rn = OXn,pn for all n. We then have an infinite sequence of completions of quadratic transforms of local rings

R0 → R1 → · · · → Rn → · · · .

Suppose that (x, y) are regular parameters in R0 = OˆX,p, and f = 0 is a local equation of C in R0 = K[[x, y]]. f is reduced by Theorem 3.6. After a linear change of variables in (x, y), we may assume that ν(f(0, y)) = r. By the Weierstrass preparation theorem, r r−1 f = u(y + a1(x)y + ··· + ar(x)), where u is a unit series. We now “complete the r-th power of f”. Set a (x) (3.3) y = y + 1 . r Then r r−1 r r−2 (3.4) y + a1(x)y + ··· + ar(x) = y + b2(x)y + ··· + br(x) for some series bi(x). We may thus assume that r r−2 (3.5) f = y + a2(x)y + ··· + ar(x). 28 3. Curve Singularities

Since f is reduced, we must have some ai(x) 6= 0. Set mult(a (x)) (3.6) n = min i . i We have n ≥ 1, since ν(f) = r. ˆ OX1,p1 has regular parameters x1, y1 such that

1. x = x1, y = x1(y1 + α) with α ∈ K, or

2. x = x1y1, y = y1 In case 2, r f = y1f1, where a2(x1y1) ar(x1y1) f1 = 1 + 2 + ··· + r y1 y1 is a local equation of the strict transform C1 of C at p1. f1 is a unit, so that

νp1 (C1) = 0. Thus case 2 cannot occur. In case 1, r f = x1f1 where r a2(x1) r−2 ar(x1) f1 = (y1 + α) + 2 (y1 + α) + ··· + r . x1 x1 f1 = 0 is a local equation at p1 for C1. If α 6= 0, then νp1 (C1) ≤ r − 1, so this case cannot occur.

If α = 0, and νp1 (C1) = r, then f1 has the form of (3.5), with n decreased by 1.

Thus for some index i with i ≤ n we have that pi must have multiplicity < r on Ci, a contradiction to the assumption that (3.2) has infinite length. The transformation of (3.3) is called a Tschirnhausen transformation (Definition 3.8). The Tschirnhausen transformation finds a (formal) curve of maximal contact H = V (y) for C at p. The Tschirnhausen transformation was introduced as a fundamental method in resolution of singularities by Abhyankar. Definition 3.12. Suppose that C is a curve on a non-singular surface S over a field K, and p ∈ C. A non-singular curve H = V (g) ⊂ spec(OˆS,p) is a formal curve of maximal contact for C at p if whenever

Sn → Sn−1 → · · · → S1 → spec(OˆS,p) is a sequence of blow-ups of points pi ∈ Si, such that the strict transform

Ci of C in Si contains pi with νpi (Ci) = νp(C) for i ≤ n, then the strict transform Hi of H in Si contains pi. 3.5. Resolution of curves embedded in a non-singular surface II 29

Exercise 3.13. 1. Resolve the singularities by blowing up: a. x2 = x4 + y4, b. xy = x6 + y6, c. x3 = y2 + x4 + y4, d. x2y + xy3 = x4 + y4, e. y2 = xn, f. y4 − 2x3y2 − 4x5y + x6 − x7. 2. Suppose that D is an eﬀective divisor on a non-singular surface S. That is, there exist curves Ci on S and numbers ri ∈ N such that I = Ir1 ···Irn . D has simple normal crossings (SNCs) on S if D C1 Cn for every p ∈ S there exist regular parameters (x, y) in OS,p such that if f = 0 is a local equation for D at p, then f = unit xayb in OS,p. Find a sequence of blow-ups of points making the total transform of y2 − x3 = 0 a SNCs divisor. 3. Suppose that D is an eﬀective divisor on a non-singular surface S. Show that there exists a sequence of blow-ups of points

Sn → · · · → S such that the total transform π∗(D) is a SNCs divisor.

3.5. Resolution of curves embedded in a non-singular surface II

In this section we consider curves embedded in a non-singular surface, defined over an arbitrary field K. Lemma 3.14. Suppose that K is a field, S is a non-singular surface over K, C is a curve on S and p ∈ C is a closed point. Suppose that π : B = B(p) → S is the blow-up of p, C˜ is the strict transform of C on B and q ∈ π−1(p) ∩ C˜. Then

νq(C˜) ≤ νp(C), and if

νq(C˜) = νp(C), then K(p) = K(q).

Proof. Let R = OˆS,p,(x, y) be regular parameters in R. Since there is a −1 natural embedding of π (p) in B ×S spec(R), it follows that x y q ∈ B × spec(R) = spec(R[ ]) ∪ spec(R[ ]). S y x 30 3. Curve Singularities

y ∼ Without loss of generality, we may assume that q ∈ spec(R[ x ]). Let K1 = y y K(p) be a coeﬃcient ﬁeld of R, m ⊂ R[ x ] the ideal of q. Since R[ x ]/(x, y) ∼ y = K1[ x ], there exists an irreducible monic polynomial h(t) ∈ K1[t] such that y y mR[ ] = (x, h( )). x x Let f ∈ R be such that f = 0 is a local equation of C, and let r = νp(C). We have an expansion in R, X i j f = aijx y i+j≥r r with aij ∈ K1. Thus f = x f1, where f1 = 0 is a local equation of the strict ˜ y transform C of C in spec(R[ x ]) and X y f = a ( )j + xΩ 1 ij x i+j=r y ˜ in R[ x ]. Let n = νq(C). Then y X y h( )n divides a ( )j x ij x i+j=r y in K1[ x ]. Also, X y deg( a ( )j) ≤ r ij x i+j=r y ˜ y in K1[ x ] implies n ≤ r and νq(C) = r implies deg(h( x )) = 1, so that y h = x − α with α ∈ K1 and (by Lemma 3.5) there exist regular parameters (x1, y1) in mq ⊂ OB(p),q such that ˆ ˆ OS,p = K1[[x, y]] → K1[[x1, y1]] = OB(p),q ∼ is the natural K1 = K(p) algebra homomorphism

x = x1, y = x1(y1 + α). Theorem 3.15. Suppose that C is a curve which is a subvariety of a non- singular surface X over an inﬁnite ﬁeld K. Then there exists a sequence of blow-ups of points λ : Y → X such that the strict transform C˜ of C on Y is non-singular.

Proof. Let r = max{νp(C) | p ∈ C}. If r = 1, C is non-singular, so we may assume that r > 1. The set {p ∈ C | νp(C) = r} is a subset of the singular locus of C, which is a proper closed subset of the 1-dimensional variety C, so it is a ﬁnite set. The proof is by induction on r. 3.5. Resolution of curves embedded in a non-singular surface II 31

We can construct a sequence of projective morphisms

πn π1 (3.7) · · · → Xn → · · · → X1 → X0 = X, where each πn+1 : Xn+1 → Xn is the blow-up of all points on the strict transform Cn of C which have multiplicity r on Cn. If this sequence is finite, then there is an integer n such that all points on the strict transform Cn of C have multiplicity ≤ r − 1. By induction on r, we can then repeat this process to construct the desired morphism Y → X which induces a resolution of C. We will assume that the sequence (3.7) is infinite, and derive a contradiction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cn which have multiplicity r on Cn and are such that πn+1(pn+1) = pn. Let ˆ Rn = OXn,pn for all n. We then have an infinite sequence of completions of quadratic transforms (blow-ups of maximal ideals) of local rings

R = R0 → R1 → · · · → Rn → · · · .

We will deﬁne 1 δ ∈ pi r!N such that

(3.8) δpi = δpi−1 − 1 for all i ≥ 1. We can thus conclude that pi has multiplicity < r on the strict transform Ci of C for some natural number i ≤ δp + 1. From this contradiction it will follow that (3.7) is a sequence of ﬁnite length.

Suppose that f ∈ R0 = OˆX,p is such that f = 0 is a local equation of C and (x, y) are regular parameters in R = OˆX,p such that r = mult(f(0, y)). We will call such (x, y) good parameters for f. Let K0 be a coeﬃcient ﬁeld of R. There is an expansion

X i j f = aijx y i+j≥r

0 with aij ∈ K for all i, j and a0r 6= 0. Deﬁne i (3.9) δ(f; x, y) = min | j < r and a 6= 0 . r − j ij We have δ(f; x, y) ≥ 1 since (x, y) are good parameters We thus have an expression (with δ = δ(f; x, y))

X i j X i j f = aijx y = Lδ + aijx y , i+jδ≥rδ i+jδ>rδ 32 3. Curve Singularities where X i j r (3.10) Lδ = aijx y = a0ry + Λ i+jδ=rδ is such that a0r 6= 0 and Λ is not zero. Suppose that (x, y) are ﬁxed good parameters of f. Deﬁne (3.11) n X 1 δ = sup{δ(f; x, y ) | y = y + b xi with n ∈ and b ∈ K0} ∈ ∪{∞}. p 1 1 i N i r!N i=1

We cannot have δp = ∞, since then there would exist a series ∞ X i y = y1 + bix i=1 such that δ(f; x, y1) = ∞, and thus there would be a unit series γ in R such that r f = γy1. But then r = 1 since f is reduced in R, a contradiction. We see then that 1 δp ∈ r! N. After possibly making a substitution n X i y = y1 + bix i=1 0 with bi ∈ K , we may assume that δp = δ(f; x, y).

Let δ = δp.

Since νp1 (C1) = r, by Lemma 3.14, we have an expression 0 R1 = K [[x1, y1]], 0 where R → R1 is the natural K -algebra homomorphism such that either

x = x1, y = x1(y1 + α) for some α ∈ K0, or

x = x1y1, y = y1.

We ﬁrst consider the case where x = x1y1, y = y1. A local equation of C1 (in spec(R1)) is f1 = 0, where f X f = = a xi + y Ω = a + x g + y h 1 yr ij 1 1 0r 1 1 1 i+j=r for some Ω, g, h ∈ R1. Thus f1 is a unit in R1, a contradiction. 3.5. Resolution of curves embedded in a non-singular surface II 33

0 Now consider the case where x = x1, y = x1(y1 + α) with 0 6= α ∈ K . A local equation of C1 is f X f = = a (y + α)j + x Ω 1 xr ij 1 1 1 i+j=r for some Ω ∈ R1. Since νp1 (C1) = r, X j r aij(y1 + α) = a0ry1. i+j=r

Substituting t = y1 + α, we have X j r aijt = a0r(t − α) . i+j=r y r If we now substitute t = x and multiply the series by x , we obtain the leading form L of f, X i j r r r r r L = aijx y = a0r(y − αx) = a0ry + ··· + (−1) a0rα x . i+j=r

Comparing with (3.10), we see that r = rδ and δ = 1, so that Lδ = L. But we can replace y with y−αx to increase δ, a contradiction to the maximality of δ. Thus νp1 (C1) < r, a contradiction. 0 Finally, consider the case x = x1, y = x1y1. Set δ = δ − 1. A local equation of C1 is f X i+j−r j f1 = r = aijx y x1 1 1 i+jδ≥rδ X i j X i j = ai−j+r,jx1y1 + ai−j+r,jx1y1, i+jδ0=rδ0 i+jδ0>rδ0 where i = i + j − r. Since 1 Lδ0 (x1, y1) = r Lδ(x1, x1y1) x1 0 has at least two non-zero terms, we see that δ(f1; x1, y1) = δ = δ − 1.

We will now show that δp1 = δ(f1; x1, y1). Suppose not. Then we can make a substitution 0 X i 0 d y1 = y1 − bix1 = y1 − bx1 + higher order terms in x1 with 0 6= b ∈ K0 such that 0 δ(f1; x1, y1) > δ(f1; x1, y1). Then we have an expression X i 0 d j 0 r X i 0 j ai−j+r,jx1(y1 − bx1) = a0r(y1) + bijx1(y1) , i+jδ0=rδ0 i+jδ0>rδ0 34 3. Curve Singularities

0 so that δ = d ∈ N, and X i 0 d j 0 r ai−j+r,jx1(y1 − bx1) = a0r(y1) . i+jδ0=rδ0 Thus X i+j−r j X i j δ0 r aijx1 y1 = ai−j+r,jx1y1 = a0r(y1 + bx1 ) . i+jδ=rδ i+jδ0=rδ0 r If we now multiply these series by x1, we obtain X i j δ r Lδ = aijx y = a0r(y + bx ) . i+jδ=rδ But we can now make the substitution y0 = y − bxδ and see that 0 δ(f; x, y ) > δ(f; x, y) = δp, 0 a contradiction, from which we conclude that δp1 = δ = δp − 1. We can then inductively deﬁne δpi for i ≥ 0 by this procedure so that (3.8) holds. The conclusions of the theorem now follow. Remark 3.16. 1. This proof is a generalization of the algorithm of Section 2.1. A more general version of this, valid in arbitrary two-dimensional regular local rings, can be found in [5] or [73].

2. Good parameters (x, y) for f which achieve δp = δ(f; x, y) are such that y = 0 is a (formal) curve of maximal contact for C at p (Deﬁ- nition 3.12). Exercise 3.17.

1. Prove that the δp deﬁned in formula (3.11) is equal to

δp = sup{δ(f; x, y) | (x, y) are good parameters for f}.

Thus δp does not depend on the initial choice of good parameters, and δp is an invariant of p.

2. Suppose that νp(C) > 1. Let π : B(p) → X be the blow-up of p, C˜ be the strict transform of C. Show that there is at most one point −1 q ∈ π (p) such that νq(C˜) = νp(C). 3. The Newton polygon N(f; x, y) is defined as follows. Let I be the ideal in R = K[[x, y]] generated by the monomials xαyβ such that α β P α β the coefficient aαβ of x y in f(x, y) = aαβx y is not zero. Set 2 α β P (f; x, y) = {(α, β) ∈ Z |x y ∈ I}. 2 Now define N(f; x, y) to be the smallest convex subset of R such that N(f; x, y) contains P (f; x, y) and if (α, β) ∈ N(f; x, y), (s, t) ∈ 3.5. Resolution of curves embedded in a non-singular surface II 35

2 R+, then (α + s, β + t) ∈ N(f; x, y). Now suppose that (x, y) are good parameters for f (so that ν(f(0, y)) = ν(f) = r). Then (0, r) ∈ N(f; x, y). Let the slope of the steepest segment of N(f; x, y) be s(f; x, y). We have 0 ≥ s(f; x, y) ≥ −1, since aαβ = 0 if α + β < r. Show that 1 δ(f; x, y) = − . s(f; x, y)

Chapter 4

Resolution Type Theorems

4.1. Blow-ups of ideals

Suppose that X is a variety, and J ⊂ OX is an ideal sheaf. The blow-up of J is M π : B = B(J ) = proj( J n) → X. n≥0 B is a variety and π is proper. If X is projective then B is projective. π is an isomorphism over X − V (J ), and JOB is a locally principal ideal sheaf.

If U ⊂ X is an open aﬃne subset, and R = Γ(U, OX ), I = Γ(U, J ) = (f1, . . . , fm) ⊂ R, then M π−1(U) = B(I) = proj( In). n≥0 If X is an integral scheme, we have

m M [ f1 fm proj( In) = spec(R[ , ··· , ]). fi fi n≥0 i=1

Suppose that W ⊂ X is a subscheme with ideal sheaf IW on X. The total transform of W , π∗(W ), is the subscheme of B with the ideal sheaf

Iπ∗(W ) = IW OB. Set-theoretically, π∗(W ) is the preimage of W , π−1(W ).

37 38 4. Resolution Type Theorems

Let U = X − V (J ). The strict transform W˜ of W is the Zariski closure of π−1(W ∩ U) in B(J ). Lemma 4.1. Suppose that R is a Noetherian ring, I,J ⊂ R are ideals and √ I = q1 ∩ · · · ∩ qm is a primary decomposition, with primes pi = qi. Then ∞ [ n \ (I : J ) = qi.

n=0 {i|J6⊂pi} The proof of Lemma 4.1 follows easily from the deﬁnition of a primary S∞ n ideal. Geometrically, Lemma 4.1 says that n=0(I : J ) removes the primary components qi of I such that V (qi) ⊂ V (J). Thus we see that the strict transform W˜ of W has the ideal sheaf ∞ [ n IW˜ = (IW OB : J OB). n=0 For q ∈ B, n IW˜ ,q = {f ∈ OB,q | fJq ⊂ IW OB,q for some n ≥ 0}. The strict transform has the property that M n W˜ = B(JOW ) = proj( (JOW ) ) n≥0 This is shown in Corollary II.7.15 [47]. Theorem 4.2 (Universal Property of Blowing Up). Suppose that I is an ideal sheaf on a variety V and f : W → V is a morphism of varieties such that IOW is locally principal. Then there is a unique morphism g : W → B(I) such that f = π ◦ g.

This is proved in Proposition II.7.14 [47]. Theorem 4.3. Suppose that C is an integral curve over a ﬁeld K. Consider the sequence

πn π1 (4.1) · · · → Cn → · · · → C1 → C, where Cn+1 → Cn is obtained by blowing up the (ﬁnitely many) singular points on Cn. Then this sequence is ﬁnite. That is, there exists n such that Cn is non-singular.

Proof. Without loss of generality, C = spec(R) is aﬃne. Let R be the integral closure of R in the function ﬁeld K(C) of C. R is a regular ring. Let C = spec(R). All ideals in R are locally principal (Proposition 9.2, page 94 [13]). By Theorem 4.2, we have a factorization

C → Cn → · · · → C1 → C 4.1. Blow-ups of ideals 39

for all n. Since C → C is finite, Ci+1 → Ci is finite for all i, and there exist affine rings Ri such that Ci = spec(Ri). For all n we have sequences of inclusions R → R1 → · · · → Rn → R.

Here Ri 6= Ri+1 for all i, since a maximal ideal m in Ri is locally principal if and only if (Ri)m is a regular local ring. Since R is finite over R, we have that (4.1) is finite. Corollary 4.4. Suppose that X is a variety over a field K and C is an integral curve on X. Consider the sequence

πn π1 · · · → Xn → Xn−1 → · · · → X1 → X, where πi+1 is the blow-up of all points of X which are singular on the strict transform Ci of C on Xi. Then this sequence is ﬁnite. That is, there exists an n such that the strict transform Cn of C is non-singular.

Proof. The induced sequence

· · · → Cn → · · · → C1 → C of blow-ups of points on the strict transform Ci of C is ﬁnite by Theorem 4.3. Theorem 4.5. Suppose that V and W are varieties and V → W is a projective birational morphism. Then V =∼ B(I) for some ideal sheaf I ⊂ OW .

This is proved in Proposition II.7.17 [47]. Suppose that Y is a non-singular subvariety of a variety X. The monoidal transform of X with center Y is π : B(IY ) → X. We define the exceptional divisor of the monoidal transform π (with center Y ) to be the reduced divisor with the same support as π∗(Y ). In general, we will define the exceptional locus of a proper birational morpism φ : V → W to be the (reduced) closed subset of V on which φ is not an isomorphism. Remark 4.6. If Y has codimension greater than or equal to 2 in X, then the exceptional divisor of the monoidal transform π : B(IY ) → X with center Y , is the exceptional locus of π. However, if X is singular and not locally factorial, it may be possible to blow up a non-singular codimension 1 subvariety Y of X to get a monoidal transform π such that the exceptional locus of π is a proper closed subset of the exceptional divisor of π. For an example of this kind, see Exercise 4.7 at the end of this section. If X is non-singular and Y has codimension 1 in X, then π is an isomorphism and 40 4. Resolution Type Theorems we have defined the exceptional divisor to be Y . This property will be useful in our general proof of resolution in Chapter 6.

Exercise 4.7. 1. Let K be an algebraically closed ﬁeld, and X be the aﬃne surface

X = spec(K[x, y, z]/(xy − z2)).

Let Y = V (x, z) ⊂ X, and let π : B → X be the monoidal transform centered at the non-singular curve Y . Show that π is a resolution of singularities. Compute the exceptional locus of π, and compute the exceptional divisor of the monoidal transform π of Y . 2. Let K be an algebraically closed ﬁeld, and X be the aﬃne 3-fold

X = spec(K[x, y, z, w]/(xy − zw).

Show that the monodial transform π : B → X with center Y is a resolution of singularities in each of the following cases, and describe the exceptional locus and exceptional divisor (of the monoidal transform): a. Y = V ((x, y, z, w)), b. Y = V ((x, z)), c. Y = V ((y, z)). Show that the monoidal transforms of cases b and c factor the monoidal transform of case a.

4.2. Resolution type theorems and corollaries

Resolution of singularities. Suppose that V is a variety. A resolution of singularities of V is a proper birational morphism φ : W → V such that W is non-singular.

Principalization of ideals. Suppose that V is a non-singular variety, and I ⊂ OV is an ideal sheaf. A principalization of I is a proper birational morphism φ : W → V such that W is non-singular and IOW is locally principal.

Suppose that X is a non-singular variety, and I ⊂ OX is a locally principal ideal. We say that I has simple normal crossings (SNCs) at p ∈ X if there exist regular parameters (x1, . . . , xn) in OX,p such that Ip = a1 an x1 ··· xn OX,p for some natural numbers a1, . . . , an. 4.2. Resolution type theorems and corollaries 41

Suppose that D is an eﬀective divisor on X. That is, D = r1E1 + ··· + rnEn, where Ei are irreducible codimension 1 subvarities of X, and ri are natural numbers. D has SNCs if I = Ir1 ···Irn has SNCs. D E1 En Suppose that W ⊂ X is a subscheme, and D is a divisor on X. We say that W has SNCs with D at p if 1. W is non-singular at p, 2. D is a SNC divisor at p,

3. there exist regular parameters (x1, . . . , xn) in OX,p and r ≤ n such that IW,p = (x1, . . . , xr) (or IW,p = OX,p) and

a1 an ID,p = x1 ··· xn OX,p

for some ai ∈ N.

Embedded resolution. Suppose that W is a subvariety of a non-singular variety X. An embedded resolution of W is a proper morphism π : Y → X which is a product of monoidal transforms, such that π is an isomorphism on an open set which intersects every component of W properly, the exceptional divisor of π is a SNC divisor D, and the strict transform W˜ of W has SNCs with D.

Resolution of indeterminacy. Suppose that φ : W → V is a rational map of proper K-varieties such that W is non-singular. A resolution of indeterminacy of φ is a proper non-singular K-variety X with a birational morphism ψ : X → W and a morphism λ : X → V such that λ = φ ◦ ψ.

Lemma 4.8. Suppose that resolution of singularities is true for K-varieties of dimension n. Then resolution of indeterminacy is true for rational maps from K-varieties of dimension n.

Proof. Let φ : W → V be a rational map of proper K-varieties, where W is non-singular. Let U be a dense open set of W on which φ is a morphism. Let Γφ be the Zariski closure of the image in W ×V of the map U → W ×V deﬁned by p 7→ (p, φ(p)). By resolution of singularities, there is a proper birational morphism X → Γφ such that X is non-singular. Theorem 4.9. Suppose that K is a perfect ﬁeld, resolution of singularities is true for projective hypersurfaces over K of dimension n, and principalization of ideals is true for non-singular varieties of dimension n over K. Then resolution of singularities is true for projective K-varieties of dimension n. 42 4. Resolution Type Theorems

Proof. Suppose V is an n-dimensional projective K-variety. If V1,...,Vr are the irreducible components of V , we have a projective birational morphism from the disjoint union of the Vi to V . Thus it suffices to assume that V is irreducible. The function field K(V ) of V is a finite separable extension of a rational function field K(x1, . . . , xn) (Chapter II, Theorem 30, page 104 [92]). By the theorem of the primitive element, ∼ K(V ) = K(x1, . . . , xn)[xn+1]/(f). We can clear the denominator of f so that

X i1 in+1 f = ai1...in+1 x1 ··· xn+1 is irreducible in K[x1, . . . , xn+1]. Let

d = max{i1 + ··· in+1 | ai1...in+1 6= 0}. Set X d−(i1+···+in+1) i1 in+1 F = ai1...in+1 X0 X1 ··· Xn+1 , n+1 the homogenization of f. Let V be the variety defined by F = 0 in P . Then K(V ) =∼ K(V ) implies there is a birational rational map from V to V . That is, there is a birational morphism φ : V˜ → V , where V˜ is a dense open subset of V . Let Γφ be the Zariski closure of {(a, φ(a)) | a ∈ V˜ } in V × V . We have birational projection morphisms π1 :Γφ → V and π2 :Γφ → V .Γφ is the blow-up of an ideal sheaf J on V (Theorem 4.5). By resolution of singularities for n-dimensional hypersurfaces, we have a resolution of singularities f : W 0 → V . By principalization of ideals in non- singular varieties of dimension n, we have a principalization g : W → W 0 for JOW 0 . By the universal property of blowing up (Theorem 4.2), we have a morphism h : W → Γφ. Hence π1 ◦ h : W → V is a resolution of singularities. Corollary 4.10. Suppose that C is a projective curve over an infinite perfect field K. Then C has a resolution of singularities.

Proof. By Theorem 3.15, resolution of singularities is true for projective plane curves over K. All ideal sheaves on a non-singular curve are locally principal, since the local rings of points are Dedekind local rings. Note that Theorem 4.3 and Corollary 4.4 are stronger results than Corol- lary 4.10. However, the ideas of the proofs of resolution of plane curves in Theorems 3.11 and 3.15 extend to higher dimensions, while Theorem 4.3 does not. As a consequence of embedded resolution of curve singularities on a surface, we will prove principalization of ideals in non-singular varieties of dimension two. This simple proof is by Abhyankar (Proposition 6, [73]). 4.2. Resolution type theorems and corollaries 43

Theorem 4.11. Suppose that S is a non-singular surface over a ﬁeld K, and J ⊂ OS is an ideal sheaf. Then there exists a ﬁnite sequence of blow-ups of points T → S such that JOT is locally principal.

Proof. Without loss of generality, S is aﬃne. Let

J = Γ(J , OS) = (f1, . . . , fm).

By embedded resolution√ of curve singularities on a surface (Exercise 3.13 of Section 3.4) applied to f1 ··· fm ∈ S, there exists a sequence of blow-ups of points π1 : S1 → S such that f1 ··· fm = 0 is a SNC divisor everywhere on S1.

Let {p1, . . . , pr} be the ﬁnitely many points on S1 where JOS is not p1 locally principal (they are contained in the singular points of V ( JOS1 )). Suppose that p ∈ {p1, . . . , pr}. By induction on the number of generators of

J, we may assume that JOS1,p = (f, g). After possibly multiplying f and g by units in OS1,p, there are regular parameters (x, y) in OS1,p such that f = xayb, g = xcyd.

Set tp = (a − c)(b − d). (f, g) is principal if and only if tp ≥ 0. By our assumption, tp < 0. Let π2 : S2 → S1 be the blow-up of p, and suppose that −1 q ∈ π (p). We will show that tq > tp. The only two points q where (f, g) may not be principal have regular parameters (x1, y1) such that

x = x1, y = x1y1 or x = x1y1, y = y1.

If x = x1, y = x1y1, then a+b b c+d d f = x1 y1, g = x1 y1, 2 tq = (a + b − (c + d))(b − d) = (b − d) + (a − c)(b − d) > tp. The case when x = x1y1, y = y1 is similar.

By induction on min{tp | JOS1,p is not principal} we can principalize J after a ﬁnite number of blow-ups of points.

Chapter 5

Surface Singularities

5.1. Resolution of surface singularities

In this section, we give a simple proof of resolution of surface singularities in characteristic zero. The proof is by the good point algorithm of Abhyankar ([7], [62], [73]). Theorem 5.1. Suppose that S is a projective surface over an algebraically closed ﬁeld K of characteristic 0. Then there exists a resolution of singularities Λ: T → S.

Theorem 5.1 is a consequence of Theorems 5.2, 4.9 and 4.11. Theorem 5.2. Suppose that S is a hypersurface of dimension 2 in a non- singular variety V of dimension 3, over an algebraically closed ﬁeld K of characteristic 0. Then there exists a sequence of blow-ups of points and non-singular curves contained in the strict transform Si of S,

Vn → Vn−1 → · · · → V1 → V, such that the strict transform Sn of S on Vn is non-singular.

The remainder of this section will be devoted to the proof of Theorem 5.2. Suppose that V is a non-singular three-dimensional variety over an algebraically closed field K of characteristic 0, and S ⊂ V is a surface. For a natural number t, define (Definitions A.17 and A.20)

Singt(S) = {p ∈ V | νp(S) ≥ t}.

By Theorem A.19, Singt(S) is Zariski closed in V .

45 46 5. Surface Singularities

Let r = max{t | Singt(S) 6= ∅} be the maximal multiplicity of points of S. There are two types of blow-ups of non-singular subvarieties on a non-singular three-dimensional variety, a blow-up of a point, and a blow-up of a non-singular curve. We will ﬁrst consider the blow-up π : B(p) → V of a closed point p ∈ V . Suppose that U = spec(R) ⊂ V is an aﬃne open neighborhood of p, and p has ideal mp = (x, y, z) ⊂ R. Then M y z x z x y π−1(U) = proj( mn) = spec(R[ , ]) ∪ spec(R[ , ]) ∪ spec(R[ , ]). p x x y y z z −1 ∼ 2 The exceptional divisor is E = π (p) = P . −1 At each closed point q ∈ π (p), we have regular parameters (x1, y1, z1) of the following forms:

x = x1, y = x1(y1 + α), z = x1(z1 + β), with α, β ∈ K, x1 = 0 a local equation of E, or

x = x1y1, y = y1, z = y1(z1 + α) with α ∈ K, y1 = 0 a local equation of E, or

x = x1z1, y = y1z1, z = z1, z1 = 0 a local equation of E. We will now consider the blow-up π : B(C) → V of a non-singular curve C ⊂ V . If p ∈ V and U = spec(R) ⊂ V is an open aﬃne neighborhood of p in V such that mp = (x, y, z) and the ideal of C is I = (x, y) in R, then M x y π−1(U) = proj( In) = spec(R[ ]) ∪ spec(R[ ]). y x −1 ∼ 1 −1 ∼ 1 −1 Also, π (p) = P and π (C ∩ U) = (C ∩ U) × P . Let E = π (C) be the exceptional divisor. E is a projective bundle over C. At each point −1 q ∈ π (p), we have regular parameters (x1, y1, z1) such that

x = x1, y = x1(y1 + α), z = z1, where α ∈ K, x1 = 0 is a local equation of E, or

x = x1y1, y = y1, z = z1, where y1 = 0 is a local equation of E. In this section, we will analyze the blow-up

π : B(W ) = B(IW ) → V of a non-singular subvariety W of V above a closed point p ∈ V , by passing to a formal neighborhood spec(OˆV,p) of p and analyzing the map ∼ π : B(IˆW,p) → spec(OˆV,p). We have a natural isomorphism B(IˆW,p) = 5.1. Resolution of surface singularities 47

B(IW ) ×V spec(OˆV,p). Observe that we have a natural identiﬁcation of π−1(p) with π−1(p).

We say that an ideal I ⊂ OˆV,p is algebraic if there exists an ideal J ⊂ OV,p such that I = Jˆ. This is equivalent to the statement that there exists an ideal sheaf I ⊂ OV such that Iˆp = I.

If I ⊂ OˆV,p is algebraic, so that there exists an ideal sheaf I ⊂ OV such that Iˆp = I, then we can extend the blow-up π : B(I) → spec(OˆV,p) to a blow-up π : B(I) → V .

The maximal ideal mpOˆV is always algebraic. However, the ideal sheaf I of a non-singular (formal) curve in spec(OˆV,p) may not be algebraic. One example of a formal, non-algebraic curve is x ˆ I = (y − e ) ⊂ C[[x, y]] = O 2 ,0. AC A more subtle example, which could occur in the course of this section, is the ideal sheaf of the irreducible curve J = (y2 − x2 − x3, z) ⊂ R = K[x, y, z], which we studied after Theorem 3.6. In Rˆ = K[[x, y, z]] we have regular parameters √ √ x = y − x 1 + x, y = y + x 1 + x, z = z. Thus JRˆ = (xy, z) = (x, z) ∩ (y, z) ⊂ Rˆ = K[[x, y, z]]. In this example, we may be tempted to blow-up one of the two formal branches x = 0, z = 0 or y = 0, z = 0, but the resulting blown-up scheme 3 will not extend to a blow-up of an ideal sheaf in AK . A situation which will arise in this section when we will blow up a formal curve which will actually be algebraic is given in the following lemma. Lemma 5.3. Suppose that V is a non-singular three-dimensional variety, p ∈ V is a closed point and π : V1 = B(p) → V is the blow-up of p with ex- −1 ∼ 2 ceptional divisor E = π (p) = P . Let R = OV,p. We have a commutative diagram of morphisms of schemes

B = B(mpRˆ) → V1 = B(p) π ↓ ↓ π spec(Rˆ) → V

−1 −1 such that π (mp) → π (p) = E is an isomorphism of schemes. Suppose that I ⊂ Rˆ is any ideal, and I˜ ⊂ OB is the strict transform of I. Then there exists an ideal sheaf J on V1 such that JOB = IEOB + I˜. 48 5. Surface Singularities

˜ Proof. IOE is an ideal sheaf on E, so there exists an ideal sheaf J ⊂ OV1 ∼ ˜ such that IE ⊂ J and J /IE = IOE. Thus J has the desired property. Lemma 5.4. Suppose that V is a non-singular three-dimensional variety, S ⊂ V is a surface, C ⊂ Singr(S) is a non-singular curve, π : B(C) → V is the blow-up of C, and S˜ is the strict transform of S in B(C). Suppose −1 that p ∈ C. Then νq(S˜) ≤ r for all q ∈ π (p), and there exists at most one −1 −1 point q ∈ π (p) such that νq(S˜) = r. In particular, if E = π (C), then ˜ either Singr(S) ∩ E is a non-singular curve which maps isomorphically onto ˜ C, or Singr(S) ∩ E is a ﬁnite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausen transformation (Deﬁnition 3.8), a local equation f = 0 of S in OˆS,p = K[[x, y, z]] has the form

r r−2 (5.1) f = z + a2(x, y)z + ··· + ar(x, y). (r) ˆr ∂f ˆr−1 ∂r−1f ˆ But f ∈ (IC,p)ˆ= IC,p implies ∂z ∈ IC,p , and r!z = ∂zr−1 ∈ IC,p. Thus z ∈ IˆC,p. After a change of variables in x and y, we may assume that ˆ ˆr i −1 IC,p = (x, z). Then f ∈ IC,p implies x | ai for all i. If q ∈ π (p), then ˆ OB(C),q has regular parameters (x1, y, z1) such that

x = x1z1, z = z1 or

x = x1, z = x1(z1 + α) for some α ∈ K. In the ﬁrst case, a local equation of the strict transform of S is a unit. In the second case, the strict transform of S has a local equation

r a2 r−2 ar f1 = (z1 + α) + 2 (z1 + α) + ··· + r . x1 x1

We have ν(f1) ≤ r, and ν(f1) < r if α 6= 0.

Lemma 5.5. Suppose that p ∈ Singr(S) is a point, π : B(p) → V is the blow-up of p, S˜ is the strict transform of S in B(p), and E = π−1(p). Then ˜ −1 ˜ νq(S) ≤ r for all q ∈ π (p), and either Singr(S)∩E is a non-singular curve ˜ or Singr(S) ∩ E is a ﬁnite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausen transformation, a local equation f = 0 of S in OˆS,p = K[[x, y, z]] has the form

r r−2 (5.2) f = z + a2(x, y)z + ··· + ar(x, y). 5.1. Resolution of surface singularities 49

−1 ˜ ˆ If q ∈ π (p) and νq(S) ≥ r, then OB(C),q has regular parameters (x1, y, z1) such that x = x1y1, y = y1, z = y1z1. or x = x1, y = x1(y1 + α), z = x1z1 ˜ ˜ for some α ∈ K, and ν1(S) = r. Thus Singr(S) ∩ E is contained in the line which is the intersection of E with the strict transform of z = 0 in ˆ B(C) ×V spec(OV,p).

Deﬁnition 5.6. Singr(S) has simple normal crossings (SNCs) if

1. all irreducible components of Singr(S) (which could be points or curves) are non-singular, and

2. if p is a singular point of Singr(S), then there exist regular parameters (x, y, z) in O such that I = (xy, z). V,p Singr(S),p

Lemma 5.7. Suppose that Singr(S) has simple normal crossings, W is a 0 point or an irreducible curve in Singr(S), π : V = B(W ) → V is the blow- 0 0 0 up of W and S is the strict transform of S on V . Then Singr(S ) has simple normal crossings.

Proof. This follows from Lemmas 5.4 and 5.5 and a simple local calculation 0 on V . Definition 5.8. A closed point p ∈ S is a pregood point if, in a neighborhood of p, Singr(S) is either empty, a non-singular curve through p, or a a union of two non-singular curves intersecting transversally at p (satisfying 2 of Definition 5.6). Definition 5.9. p ∈ S is a good point if p is pregood and for any sequence

Xn → Xn−1 → · · · → X1 → spec(OV,p) of blow-ups of non-singular curves in Singr(Si), where Si is the strict transform of S ∩ spec(OV,p) on Xi, then q is pregood for all closed points q ∈ Singr(Sn). In particular, Singr(Sn) contains no isolated points. A point which is not good is called bad.

Lemma 5.10. Suppose that all points of Singr(S) are good. Then there exists a sequence of blow ups 0 V = Vn → · · · → V1 → V of non-singular curves contained in Singr(Si), where Si is the strict trans- 0 0 form of S in Vi, such that Singr(S ) = ∅ if S is the strict transform of S on V 0. 50 5. Surface Singularities

Proof. Suppose that C is a non-singular curve in Singr(S). Let π1 : V1 = B(C) → V be the blow-up of C, and let S1 be the strict transform of S. If Singr(S1) 6= ∅, we can choose another non-singular curve C1 in Singr(S1) and blow-up by π2 : V2 = B(C1) → V1. Let S2 be the strict transform of S1. We either reach a surface Sn such that Singr(Sn) = ∅, or we obtain an inﬁnite sequence of blow-ups

· · · → Vn → Vn−1 → · · · → V such that each Vi+1 → Vi is the blow-up of a curve Ci in Singr(Si), where Si is the strict transform of S on Vi. Each curve Ci which is blown up must map onto a curve in S by Lemma 5.5. Thus there exists a curve γ ⊂ S such that there are inﬁnitely many blow-ups of curves mapping onto γ in the above sequence. Let R = OV,γ, a two-dimensional regular local ring. IS,γ is a height 1 prime ideal in this ring, and P = Iγ,γ is the maximal ideal of R. We have

dim R + trdegK R/P = 3 by the dimension formula (Theorem 15.6 [66]). Thus R/P has transcendence degree 1 over K. Let t ∈ R be the lift of a transcendence basis of R/P over K. Then K[t] ∩ P = (0), so the ﬁeld K(t) ⊂ R. We can write R = AQ, where A is a ﬁnitely generated K(t)-algebra (which is a domain) and Q is a maximal ideal in A. Thus R is the local ring of a non-singular point q on the K(t) surface spec(A). q is a point of multiplicity r on the curve in spec(A) with ideal sheaf IS,γ in R. The sequence

· · · → Vn ×V spec(R) → Vn−1 ×V spec(R) → · · · → spec(R) consists of inﬁnitely many blow-ups of points on a K(t)-surface, which are of multiplicity r on the strict transform of the curve spec(R/IS,γ). But this is impossible by Theorem 3.15. Lemma 5.11. The number of bad points on S is ﬁnite.

Proof. Let

B0 = {isolated points of Singr(S)} ∪ {singular points of Singr(S)}.

Singr(S) − B0 is a non-singular 1-dimensional subscheme of V . Let

π1 : V1 = B(Singr(S) − B0) → V − B0 be its blow-up. Let S1 be the strict transform of S, and let

B1 = {isolated points of Singr(S1)} ∪ {singular points of Singr(S1)}. We can iterate to construct a sequence

· · · → (Vn − Bn) → · · · → V, 5.1. Resolution of surface singularities 51

where πn : Vn − Bn → V − Tn are the induced surjective maps, with

Tn = B0 ∪ π1(B1) ∪ · · · ∪ πn(Bn).

Let Sn be the strict transform of S on Vn.

let C ⊂ Singr(S) be a curve. spec(OS,C ) is a curve singularity of multiplicity r, embedded in a non-singular surface over a ﬁeld of transcendence degree 1 over K (as in the proof of Lemma 5.10). πn induces by base change

Sn ×S spec(OS,C ) → spec(OS,C ), which corresponds to an open subset of a sequence of blow-ups of points over spec(OS,C ). So for n >> 0, Sn ×S spec(OS,C ) is non-singular. Thus there are no curves in Singr(Sn) which dominate C. For n >> 0 there are thus no curves in Singr(Sn) which dominate curves of Singr(S), so that Singr(Sn) ∩ (Vn − Bn) is empty for large n, and all bad points of S are contained in a ﬁnite set Tn. Theorem 5.12. Let

(5.3) · · · → Vn → Vn−1 → · · · → V1 → V be the sequence where πn : Vn → Vn−1 is the blow-up of all bad points on the strict transform Sn−1 of S. Then this sequence is ﬁnite, so that it terminates after a ﬁnite number of steps with a Vm such that all points of Singr(Sm) are good.

We now give the proof of Theorem 5.12. Suppose there is an infinite sequence of the form of (5.3). Then there is an infinite sequence of points pn ∈ Singr(Sn) such that πn(pn) = pn−1 for all n. We then have an infinite sequence of homomorphisms of local rings ˆ ˆ ˆ R0 = OV,p → R1 = OV1,p1 → · · · → Rn = OVn,pn → · · · . By the Weierstrass preparation theorem and after a Tshirnhausen transformation (Definition 3.8) there exist regular parameters (x, y, z) in R0 such that there is a local equation f = 0 for S in R0 of the form r r−2 f = z + a2(x, y)z + ··· + ar(x, y) with ν(ai(x, y)) ≥ i for all i. Since νpi (Si) = r for all i, we have regular parameters (xi, yi, zi) in Ri for all i and a local equation fi = 0 for Si such that xi−1 = xi, yi−1 = xi(yi + αi), zi−1 = xi−1zi−1 with αi ∈ K, or

xi−1 = xiyi, yi−1 = yi, zi−1 = yi−1zi−1, and r r−2 fi = zi +a ˜2,i(xi, yi)zi + ··· a˜r,i(xi, yi), 52 5. Surface Singularities where  a˜j,i−1(xi,xi(yi+αi))  xj a˜ (x , y ) = i ji i i a˜j,i−1(xiyi,yi)  or j yi for all j. Thus the sequence

K[[x, y] → K[[x1, y1]] → · · · is a sequence of blow-ups of a maximal ideal, followed by completion. In K[[xi, yi]] we have relations

cj,i dj,i aj = γj,ixi yi a˜j,i−1 for all i and 2 ≤ j ≤ r, where γji is a unit. By embedded resolution of curve Qr singularities (Exercise 3.13), there exists m0 such that j=2 aj(x, y) = 0 is a SNC divisor in Ri for all i ≥ m0. Thus for i ≥ m0 we have an expression

r a2i b2i r−2 ari bri (5.4) fi = zi + a2i(xi, yi)xi yi zi + ··· + ari(xi, yi)xi yi , where the aji are units (or zero) in Ri and aji + bji ≥ j if aji 6= 0. We observe that, by the proof of Theorem A.19 and Remark A.21, √ Iˆ = J, Singr(Si),pi where J is the ideal in Ri generated by ∂j+k+lf { i | 0 ≤ j + k + l < r}. j k l ∂xi ∂yi ∂zi

Thus Iˆ must be one of (xi, yi, zi), (xi, zi), (yi, zi) or (xiyi, zi). Singr(Si),pi √ Singr(Si) has SNCs at pi unless J = (xiyi, zi) is the completion of an −1 ∼ 2 ideal of an irreducible (singular) curve on Vi. Let E = πi (pi−1) = P . By construction, we have that xi = 0 or yi = 0 is a local equation of E at pi−1. Without loss of generality, we may suppose that xi = 0 is a local equation of E. zi = 0 is a local equation of the strict transform of z = 0 in Vi ×V spec(R), so that (xi, zi) is the completion at qi of the ideal sheaf of ˆ a subscheme of E. Thus there is a curve C in Vi such that IC,pi = (xi, zi) by Lemma 5.3. The ideal sheaf J = (I : I ) in O is such that Singr(Si) C Vi (J ∩ I ) ∼ I , so that (x y , z ) is not the completion of an ideal C pi = Singr(Si),pi i i i of an irreducible curve on Vi, and Singr(Si) has simple normal crossings at pi.

We may thus assume that m0 is suﬃcently large that Singr(Si) has SNCs everywhere on Vi for i ≥ m0.

We now make the observation that we must have bji 6= 0 and aki 6= 0 for some j and k if i ≥ m0. If we did have bji = 0 for all j (with a similar analysis if aji = 0 for all j) in (5.4), then, since √ Iˆ = J, Singr(Si),pi 5.1. Resolution of surface singularities 53

where J is the ideal in Ri generated by ∂j+k+lf { i | 0 ≤ j + k + l < r}, j k l ∂xi ∂yj ∂zi √ we have (with the condition that bji = 0 for all j) that J = (xi, zi). Since Singr(Si) has SNCs in Vi, there exists a non-singular curve C ⊂ Singr(Si) such that xi = zi = 0 are local equations of C in Ri. Let λ : W → Vi be the −1 blow-up of C. If q ∈ λ (pi), we have regular parameters (u, v, y) in OˆW,q such that xi = u, zi = u(v + β) or xi = uv, zi = v. 0 0 If f = 0 is a local equation of the strict transform S of Si in W , we have 0 that νq(f ) < r except possibly if xi = u, zi = uv. Then

0 r a2i−2 r−2 ari−r f = v + a2iu v + ··· + ariu . 0 We see that f is of the form of (5.4), with bji = 0 for all j, but a min{ ji | 2 ≤ j ≤ r} j has decreased by 1. We see, using Lemma 5.4, that after a ﬁnite number of blow-ups of non-singular curves in Singr of the strict transform of S, the multiplicity must be < r at all points above pi. Thus pi is a good point, a contradiction to our assumption. Thus bji 6= 0 and aki 6= 0 in (5.4) for some j and k if i ≥ m0.

If follows that, for i ≥ m0, we must have

xi = xi+1, yi = xi+1yi+1, zi = xi+1zi+1 or xi = xi+1yi+1, yi = yi+1, zi = yi+1zi+1. Thus in (5.4), we must have for 2 ≤ j ≤ r, a = a + b − j, b = b or (5.5) j,i+1 ji ji j,i+1 ji aj,i+1 = aji, bj,i+1 = aj,i + bji − j respectively, for all i ≥ m0. Suppose that ζ ∈ R. Then the fractional part of ζ is {ζ} = t if ζ = a + t with a ∈ Z and 0 ≤ t < 1.

Lemma 5.13. For i ≥ m0, pi is a good point on Vi if there exists j such that aji 6= 0 and 54 5. Surface Singularities

1. a a b b ji ≤ ki , ji ≤ ki j k j k for all k with 2 ≤ k ≤ r and aki 6= 0, while 2. a b ji + ji < 1. j j

Proof. Since νpi (Si) = r, aji + bji ≥ j. Condition 2 thus implies that either aji ≥ j or bji ≥ j. Without loss of generality, aji ≥ j. Then condition 1 implies that aki ≥ k for all k, and Iˆ ⊂ (x , z )R . Singr(Si),pi i i i

Since Singr(Si) has SNCs, there exists a non-singular curve C ⊂ Singr(Si) ˆ 0 such that IC,pi = (xi, zi). Let π : V = B(C) → Vi be the blow-up of C, 0 0 0 and let S be the strict transform of S on V . By Lemma 5.7, Singr(S ) has SNCs. A direct local calculation shows that there is at most one point −1 0 ˆ q ∈ π (pi) such that νq(S ) = r, and if such a point q exists, then OV 0,q has regular parameters (x0, y0, z0) such that 0 0 0 0 xi = x , yi = y , zi = x z . A local equation f 0 = 0 of S0 at q is

0 0 r 0 a2i−2 0 b2i 0 r−2 0 ari−r 0 bri f = (z ) + a2i(x ) (y ) (z ) + ··· + ari(x ) (y ) . Thus we have an expression of the form (5.4), and conditions 1 and 2 of the statement of this lemma hold for f 0. After repeating this procedure a ﬁnite number of times we will construct V → Vi such that νa(S) < r for all points aji bji a on the strict transform S of S such that a maps to pi, since j + j must drop by 1 every time we blow up. Lemma 5.14. Let a a b b δ = ji − ki ji − ki . j,k,i j k j k Then

δj,k,i+1 ≥ δjki, and if δjki < 0 then 1 δ − δ ≥ . jk,i+1 jki r4 Proof. We may assume that the ﬁrst case of (5.5) holds. Then b b 2 δ = δ + ji − ki . j,k,i+1 j,k,i j k 5.1. Resolution of surface singularities 55

bji bki If δj,k,i < 0 we must have j − k 6= 0. Thus b b 2 1 1 ji − ki ≥ ≥ . j k j2k2 r4

Corollary 5.15. There exists m1 such that i ≥ m1 implies condition 1 of Lemma 5.13 holds.

Proof. There exists m1 such that i ≥ m1 implies δjki ≥ 0 for all j, k. Then aki bki the pairs ( k , k ) with 2 ≤ k ≤ r are totally ordered by ≤, so there exists aji bji a minimal element ( j , j ). 0 Lemma 5.16. There exists an i ≥ m1 such that we have condition 2 of Lemma 5.13, a 0 b 0 { ji } + { ji } < 1. j j Proof. A calculation shows that a b ji + ji ≥ 1 j j implies a b a b 1 j,i+1 + j,i+1 ≤ ji + ji − . j j j j r 0 With the above i , pi0 is a good point, since pi0 satisfies the conditions of Lemma 5.13. Thus the conclusions of Theorem 5.12 must hold. Now we give a the proof of Theorem 5.2. Let r be the maximal multiplicity of points of S. By Theorem 5.12, there exists a finite sequence of blow-ups of points Vm → V such that all points of Singr(Sm) are good points, where Sm is the strict transform of S on Vm. By Lemma 5.10, there exists a sequence of blow-ups of non-singular curves Vn → Vm such that Singr(Sn) = ∅, where Sn is the strict transform of S on Vn. By descending induction on r we can reach the case where Sing2(Sn) = ∅, so that Sn is non-singular. Remark 5.17. Zariski discusses early approaches to resolution in [91]. Some early proofs of resolution of surface singularities are by Albanese [12], Beppo Levi [61], Jung [57] and Walker [82]. Zariski gave several proofs of resolution of algebraic surfaces over fields of characteristic zero, including the first algebraic proof [86], which we present later in Chapter 8. 56 5. Surface Singularities

Exercise 5.18. 1. Suppose that S is a surface which is a subvariety of a non-singular three-dimensional variety V over an algebraically closed ﬁeld K of characteristic 0, and p ∈ S is a closed point. a. Show that a Tschirnhausen transformation of a suitable local equation of S at p gives a formal hypersurface of maximal contact for S at p. b. Show that if f = 0 is a local equation of S at p, and (x, y, z) are regular parameters at p such that ν(f) = ν(f(0, 0, z)), ∂r−1f then ∂zr−1 = 0 is a hypersurface of maximal contact (A.20) for f = 0 in a neighborhood of the origin. 2. Follow the algorithm to reduce the multiplicity of f = zr+xayb = 0, where a + b ≥ r.

5.2. Embedded resolution of singularities

In this section we will prove the following theorem, which is an extension of the main resolution theorem, Theorem 5.2 of the previous section.

Theorem 5.19. Suppose that S is a surface which is a subvariety of a non- singular three-dimensional variety V over an algebraically closed ﬁeld K of characteristic 0. Then there exists a sequence of monoidal transforms over the singular locus of S such that the strict transform of S is non-singular, and the total transform π∗(S) is a SNC divisor.

Deﬁnition 5.20. A resolution datum R = (E0,E1, S, V ) is a 4-tuple where E0,E1,S are reduced eﬀective divisors on the non-singular three-dimensional variety V such that E = E0 ∪ E1 is a SNC divisor.

R is resolved at p ∈ S if S is non-singular at p and E ∪ S has SNCs at p. For r > 0, let

Singr(R) = {p ∈ S | νp(S) ≥ r and R is not resolved at p}.

Observe that Singr(R) = Singr(S) if r > 1, and Singr(R) is a proper Zariski closed subset of S for r ≥ 1. For p ∈ S, let

η(p) = the number of components of E1 containing p.

We have 0 ≤ η(p) ≤ 3. For r, t ∈ N, let

Singr,t(R) = {p ∈ Singr(R) | η(p) ≥ t}. 5.2. Embedded resolution of singularities 57

Singr,t(R) is a Zariski closed subset of Singr(R) (and of S). For the rest of this section we will ﬁx

r = ν(R) = ν(S, E) = max{νp(S) | p ∈ S, R is not resolved at p},

t = η(R) = the maximum number of components of E1 containing a point of Singr(R). Definition 5.21. A permissible transform of R is a monoidal transform 0 π : V → V whose center is either a point of Singr,t(R) or a non-singular curve C ⊂ Singr,t(R) such that C makes SNCs with E. Let F be the exceptional divisor of π. Then π∗(E) ∪ F is a SNC divisor. 0 0 0 0 0 0 We define the strict transform R of R to be R = (E0,E1,S ), where S is the strict transform of S, and we also define 0 ∗ E0 = π (E0)red + F, 0 0 ∗ E1 = strict transform of E1 if ν(S , π ((E0 + E1)red + F ) = ν(R), 0 0 ∗ 0 ∗ E0 = ∅,E1 = π (E0 + E1)red + F if ν(S , π ((E0 + E1)red + F ) < ν(R). Lemma 5.22. With the notation of the previous definition, ν(R0) ≤ ν(R) and ν(R0) = ν(R) implies η(R0) ≤ η(R).

The proof of Lemma 5.22 is a generalization of Lemmas 5.3 and 5.4.

Deﬁnition 5.23. p ∈ Singr,t(R) is a pregood point if the following two conditions hold:

1. In a neighborhood of p, Singr,t(R) is either a non-singular curve through p, or two non-singular curves through p intersecting transversally at p (satisfying 2 of Deﬁnition 5.6).

2. Singr,t(R) makes SNCs with E at p. That is, there exist regular parameters {x, y, z} in OV,p such that the ideal sheaf of each component of E and each curve in Singr,t(R) containing p is generated by a subset of {x, y, z} at p. p ∈ S is a good point if p is a pregood point and for any sequence

π : Xn → Xn−1 → · · · → X1 → spec(OV,p) of permissible monodial transforms centered at curves in Singr,t(Ri), where −1 Ri is the transform of R on Xi, then q is a pregood point for all q ∈ π (p).

A point p ∈ Singr,t(R) is called bad if p is not good.

Lemma 5.24. The number of bad points in Singr,t(R) is ﬁnite. 58 5. Surface Singularities

Lemma 5.25. Suppose that all points of Singr,t(R) are good. Then there exists a sequence of permissible monoidal transforms, π : V 0 → V , centered at non-singular curves contained in Singr,t(Si), where Si is the strict transform of S on the i-th monoidal transform, such that if R0 is the strict transform of R on V 0, then either ν(R0) < ν(R) or ν(R0) = ν(R) and η(R0) < η(R).

The proofs of the above two lemmas are similar to the proofs of Lemmas 5.11 and 5.10 respectively, with the use of embedded resolution of plane curves.

Theorem 5.26. Suppose that R is a resolution datum such that E0 = ∅. Let

πn π1 (5.6) · · · → Vn → Vn−1 → · · · → V0 = V be the sequence where πn is the monoidal transform centered at the union of bad points in Singr,t(Ri), where Ri is the transform of R on Vi. Then this sequence is ﬁnite, so that it terminates in a Vm such that all points of Singr,t(Rm) are good.

Proof. Suppose that (5.6) is an infinite sequence. Then there exists an infinite sequence of points pn ∈ Vn such that πn+1(pn+1) = pn for all n and ˆ pn is a bad point. Let Rn = OVn,pn for n ≥ 0. Since the sequence (5.6) is infinite, its length is larger than 1, so we may assume that t ≤ 2.

Let f = 0 be a local equation of S at p0, and let gk, 0 ≤ k ≤ t, be local equations of the components of E1 containing p0. In R0 = OˆV,p there are regular parameters (x, y, z) such that ν(f(0, 0, z)) = r and ν(gk(0, 0, z)) = 1 for all k. This follows since for each equation this is a non-trivial Zariski open condition on linear changes of variables in a set of regular parameters. By the Weierstrass preparation theorem, after multiplying f and the gk by units and performing a Tschirnhausen transformation on f, we have expressions f = zr + Pr a (x, y)zr−j, (5.7) j=2 j gk = z − φk(x, y), (or gk is a unit) in R0. z = 0 is a local equation of a formal hypersurface of maximal contact ˆ for f = 0. Thus we have regular parameters (x1, y1, z1) in R1 = OV1,p1 deﬁned by x = x1, y = x1(y1 + α), z = x1z1, with α ∈ K, or x = x1y1, y = y1, z = y1z1. 5.2. Embedded resolution of singularities 59

The strict transforms f1 of f and gk1 of gk have the form of (5.7) with a replaced by aj (or aj ), φ with φk (or φk ). After a ﬁnite number of j j j k x1 y1 x1 y1 monoidal transforms centered at points pj with ν(fj) = r, we have forms in ˆ Ri = OVi,pi for the strict transforms of f and gk: f = zr + Pr a (x , y )xaji ybji zr−j, (5.8) i i j=2 j i i i i i r cki dki gki = zk + φki(xi, yi)xi yi , such that aj, φki are either units or zero (as in the proof of Theorem 5.12). i We further have that local equations of the exceptional divisor E0 of Vi → V are one of xi = 0, yi = 0 or xiyi = 0. This last case can only occur if t < 2. Note that our assumption η(pi) = t implies gki is not a unit in Ri for 1 ≤ k ≤ t. We can further assume, by an extension of the argument in the proof of Theorem 5.12, that all irreducible curves in Singr,t(Ri) are non-singular. By Lemma 5.14, we can assume that the set a b T = ji , ji , | a 6= 0 ∪ {(c , d ) | φ 6= 0} j j j ki ki ki is totally ordered. Now by Lemma 5.16 we can further assume that a b j0i + j0i < 1, j0 j0 a b where ( j0i , j0i ) is the minimum of the ( aji , bji ). Recall that, if t > 0, j0 j0 j j Qt i i=1 gi = 0 makes SNCs with the exceptional divisor E0 of Vi → V . These conditions imply that, after possibly interchanging the gki, interchanging xi, yi and multiplying xi, yi by units in Ri, that the gki can be described by one of the following cases at pi: 1. t = 0. 2. t = 1, λ a. g1i = zi + xi , λ ≥ 1. λ µ b. g1i = zi + xi y , λ ≥ 1, µ ≥ 1. c. g1i = zi 3. t = 2, a. g1i = zi, g2i = zi + xi. b. gi1 = zi + xi, gi2 = zi + xi, where is a unit in Ri such that the residue of in the residue ﬁeld of Ri is not 1. λ µ c. gi1 = zi + xi, gi2 = zi + xi yi , where is a unit in Ri, λ ≥ 1 and λ + µ ≥ 2. If µ > 0 we can take = 1.

We can check explicitly by blowing up permissible curves that pi is a good point. In this calculation, if t = 0, this follows from the proof of 60 5. Surface Singularities

Theorem 5.12, recalling that xi = 0, yi = 0 or xiyi = 0 are local equations of E0 at pi. If t = 1, we must remember that we can only blow up a curve if it lies on g1 = 0, and if t = 2, we can only blow up the curve g1 = g2 = 0. Although we are constructing a sequence of permissible monoidal transforms ˆ over spec(OVi,pi ), this is equivalent to constructing such a sequence over spec(OVi,pi ), as all curves blown up in this calculation are actually algebraic, since all irreducible curves in Singr,t(Ri) are non-singular. We have thus found a contradiction, by which we conclude that the sequence (5.6) has ﬁnite length. The proof of Theorem 5.19 now follows easily from Theorem 5.26, Lemma 5.25 and descending induction on (r, t) ∈ N×N, with the lexicographic order. Remark 5.27. This algorithm is outlined in [73] and [62]. Exercise 5.28. Resolve the following (or at least make (ν(R), η(R)) drop in the lexicographic order):

1. R = (∅,E1,S), where S has local equation f = z = 0, and E1 is the union of two components with respective local equations g1 = z + xy, g2 = z + x; 3 7 7 2. R = (∅,E1,S), where S has local equation f = z + x y = 0 and E1 is the union of two components with respective local equations 2 3 g1 = z + xy, g2 = z + y(x + y ). Chapter 6

Resolution of Singularities in Characteristic Zero

The first proof of resolution in arbitrary dimension (over a field of characteristic zero) was by Hironaka [52]. The proof consists of a local proof, and a complex web of inductions to produce a global proof. The local proof uses sophisticated methods in commutative algebra to reduce the problem of reducing the multiplicity of the strict transform of an ideal after a permissible sequence of monoidal transforms to the problem of reducing the multiplicity of the strict transform of a hypersurface. The use of the Tschirnhausen transformation to find a hypersurface of maximal contact lies at the heart of this proof. The order ν and the invariant τ (of Chapter 7) are the principal invariants considered. De Jong’s theorem [36], referred to at the end of Chapter 7, can be refined to give a new proof of resolution of singularities of varieties of characteristic zero ([9], [16], [70]). The resulting theorem is not as strong as the classical results on resolution of singularities in characteristic zero (Section 6.8), as the resulting resolution X0 → X may not be an isomorphism over the non-singular locus of X. In recent years several different proofs of canonical resolution of singularities have been constructed (Bierstone and Milman [14], Bravo, Encinas and Villamayor [17], Encinas and Villamayor [40], [41], Encinas and Hauser [39], Moh [67], Villamayor [79], [80], and Wlodarczyk [84]). These papers

61 62 6. Resolution of Singularities in Characteristic Zero have significantly simplified Hironaka’s original proofs in [47] and [55], and represent a great advancement of the subject. In this chapter we prove the basic theorems of resolution of singularities in characteristic 0. Our proof is based on the algorithms of [40] and [41]. The final statements of resolution are proved in Section 6.8, and in Exercise 6.42 at the end of that section. Throughout this chapter, unless explicitely stated otherwise, we will assume that all varieties are over a field K of characteristic zero. Suppose that X1 and X2 are subschemes of a variety W . We will denote the reduced set-theoretic intersection of X1 and X2 by X1 ∩X2. However, we will denote the scheme-theoretic intersection of X1 and X2 by X1 · X2. X1 · X2 is the subscheme of W with ideal sheaf IX1 + IX2 .

6.1. The operator 4 and other preliminaries

Definition 6.1. Suppose that K is a field of characteristic zero, R = K[[x1, . . . , xn]] is a ring of formal power series over K, and J = (f1, . . . , fr) ⊂ R is an ideal. Define ˆ ˆ ∂fi 4(J) = 4R(J) = (f1, . . . , fr) + | 1 ≤ i ≤ r, 1 ≤ j ≤ n . ∂xj

4ˆ (J) is independent of the choice of generators fi of J and regular parameters xj in R.

Lemma 6.2. Suppose that W is a non-singular variety and J ⊂ OW is an ideal sheaf. Then there exists an ideal 4(J) = 4W (J) ⊂ OW such that ˆ 4(J)OˆW,q = 4(Jˆ) for all closed points q ∈ W .

Proof. We define 4(J) as follows. We can cover W by affine open subsets 1 U = spec(R) which satisfy the conclusions of Lemma A.11, so that ΩR/K = dy1R ⊕ · · · ⊕ dynR. If Γ(U, J) = (f1, . . . , fr), we define

∂fi Γ(U, 4(J)) = (fi, | 1 ≤ i ≤ r, 1 ≤ j ≤ n). ∂yj This deﬁnition is independent of all choices made in this expression.

Suppose that q ∈ W is a closed point, and x1, . . . , xn are regular parameters in OW,q. Let U = spec(R) be an aﬃne neighborhood of q as above. 1 Let A = OW,q. Then {dy1, . . . , dyn} and {dx1, . . . , dxn} are A-bases of ΩA/k by Lemma A.11 and Theorem A.10. Thus the determinant ∂y |( i )|(q) 6= 0 ∂xj 6.1. The operator 4 and other preliminaries 63

from which the conclusions of the lemma follow.

Given J ⊂ OW and q ∈ W , we define νJ (q) = νq(J) (Definition A.17). If r X is a subvariety of W , we denote νX = νIX . We can define 4 (J) for r ∈ N inductively, by the formula 4r(J) = 4(4r−1(J)). We define 40(J) = J.

Lemma 6.3. Suppose that W is a non-singular variety, (0) 6= J ⊂ OW is an ideal sheaf, and q ∈ W (which is not necessarily a closed point). Then:

1. νq(J) = b > 0 if and only if νq(4(J)) = b − 1. b−1 2. νq(J) = b > 0 if and only if νq(4 (J)) = 1. b−1 3. νq(J) ≥ b > 0 if and only if q ∈ V (4 (J)). 4. νJ : W → N is an upper semi-continuous function.

Proof. The lemma follows from Theorem A.19. If f : X → I is an upper semi-continuous function, we deﬁne max f to be the largest value assumed by f on X, and we deﬁne a closed subset of X, Max f = {q ∈ X | f(q) = max f}. b−1 We have Max νJ = V (4 (J)) if b = max νJ . Suppose that X is a closed subset of a n-dimensional variety W . We shall denote by (6.1) R(1)(X) ⊂ X the union of irreducible components of X which have dimension n − 1.

Lemma 6.4. Suppose that W is a non-singlar variety, J ⊂ OW is an ideal sheaf, b = max νJ and Y ⊂ Max νJ is a non-singular subvariety. Let π : W1 → W be the monoidal transform with center Y . Let D be the exceptional divisor of π. Then:

1. There is an ideal sheaf J 1 ⊂ OW1 such that −b J 1 = ID J

and ID - J 1. 2. If q ∈ W1 and p = π(q), then ν (q) ≤ ν (p). J1 J 3. max ν ≥ max ν . J J1

Proof. Suppose that q ∈ W1 and p = π(q). Let A be the Zariski closure of {q} in W1, and B the Zariski closure of {p} in W . Then A → B is proper. By upper semi-continuity of νJ , there exists a closed point x ∈ B such that −1 νJ (x) = νJ (p). Let y ∈ π (x) ∩ A be a closed point. By semi-continuity 64 6. Resolution of Singularities in Characteristic Zero of ν , we have ν (q) ≤ ν (y), so it suﬃces to prove 2 when q and p are J1 J1 J1 closed points.

By Corollary A.5, there is a regular system of parameters y1, . . . , yn in OW,p such that y1 = y2 = ··· = yr = 0 are local equations of Y at p. By Remark A.18, we may assume that K = K(p) = K(q). Then we can make a linear change of variables in y1, . . . , yn to assume that there are regular parameters y1,..., yn in OW1,q such that

y1 = y1, y2 = y1y2, . . . , yr = y1yr, yr+1 = yr+1, . . . , yn = yn. y1 = 0 is a local equation of D. By Remark A.18, if suﬃces to verify the conclusions of the theorem in the complete local rings

R = OˆW,p = K[[y1, . . . , yn] and ˆ S = OW1,q = K[[y1,..., yn]].

Y ⊂ Max νJ implies νη(J) = b, where η is the general point of the component of Y whose closure contains p. Suppose that f ∈ JR. Let t = νR(f) ≥ b. Then there is an expansion

X i1 in f = ai1,...,in y1 ··· yn in R, with ai1,...,in ∈ K and ai1,...,in = 0 if i1 + ··· + ir < t. Further, there exist i1, . . . , in with i1 + ··· + in = t such that ai1,...,in 6= 0. In S, we have t f = y1f1, where X i1+···+ir−t in f1 = ai1,...,in y1 ··· yn is such that y1 does not divide f1. Thus we have νS(f1) ≤ νR(f). In b particular, we have y1 | JS; and since there exists f ∈ R with νR(f) = b, b+1 y1 - JS. Thus −b J 1 = ID J satisﬁes 1 and ν (q) ≤ ν (p) = b. J1 J

Suppose that W is a non-singular variety and J ⊂ OW is an ideal sheaf. Suppose that Y ⊂ W is a non-singular subvariety. Let Y1,...,Ym be the distinct irreducible (connected) components of Y . Let π : W1 → W be the monoidal transform with center Y and exceptional divisor D. Let D = D1 +···+Dm, where Di are the distinct irreducible (connected) components 6.1. The operator 4 and other preliminaries 65

of D, indexed so that π(Di) = Yi. Then by an argument as in the proof of

Lemma 6.4, there exists an ideal sheaf J 1 ⊂ OW1 such that

c1 cm (6.2) JOW1 = I ···I J 1, D1 Dm where J is such that for 1 ≤ i ≤ m we have I J and c = ν (J), with 1 Di - 1 i ηi ηi the generic point of Yi. J 1 is called the weak transform of J.

There is thus a function c on W1, deﬁned by c(q) = 0 if q 6∈ D and c(q) = ci if q ∈ Di, such that

c (6.3) JOW1 = IDJ 1.

In the situation of Lemma 6.4, J 1 has properties 1 and 2 of that lemma.

With the notation of Lemma 6.4, suppose that J = IX is the ideal sheaf of a subvariety X of W . The weak transform X of X is the subscheme of W1 with ideal sheaf J 1. We see that the weak transform X of X is much easier to calculate than the strict transform X˜ of X. We have inclusions

X˜ ⊂ X ⊂ π∗(X).

Example 6.5. Suppose that X is a hypersurface on a variety W , r = max νX and Y ⊂ Max νX is a non-singular subvariety. Let π : W1 → W be the monoidal transform of W with center Y and exceptional divisor E. In this case the strict transform X˜ and the weak transform X are the same scheme, and π∗(X) = X˜ + rE.

3 Example 6.6. Let X ⊂ W = A be the nodal plane curve with ideal 2 3 3 IX = (z, y − x ) ⊂ k[x, y, z]. Let m = (x, y, z), and let π : B = B(m) → A be the blow-up of the point m. The strict transform X˜ of X is non-singular. The most interesting point in X˜ is the point q ∈ π−1(m) with regular parameters (x1, y1, z1) such that

x = x1, y = x1y1, z = x1z1.

Then 2 2 3 Iπ∗(X),q = (x1z1, x1y1 − x1),

2 2 IX,q = (z1, x1y1 − x1),

2 IX,q˜ = (z1, y1 − x1). In this example, X, X˜ and π∗(X) are all distinct. 66 6. Resolution of Singularities in Characteristic Zero

6.2. Hypersurfaces of maximal contact and induction in resolution

Suppose that W is a non-singular variety, J ⊂ OW is an ideal sheaf and b ∈ N. Deﬁne Sing(J, b) = {q ∈ W | νJ (q) ≥ b}. Sing(J, b) is Zariski closed in W by Lemma 6.3. Suppose that Y ⊂ Sing(J, b) is a non-singular (but not necessarily connected) subvariety of W . Let π1 : −1 W1 → W be the monodial transform with center Y , and let D1 = π (Y ) be the exceptional divisor.

Recall (6.3) that the weak transform J 1 of J is deﬁned by c JOW1 = JD1 J 1, where c is locally constant on connected components of D1. We have c ≥ b by upper semi-continuity of νJ .

We can now deﬁne J1 by b (6.4) JOW1 = JD1 J1, so that J = J c−bJ . 1 D1 1 Suppose that

πn π1 (6.5) Wn → Wn−1 → · · · → W1 → W is a sequence of monoidal transforms such that each πi is centered at a non-singular subvariety Yi ⊂ Sing(Ji, b), where Ji is deﬁned inductively by b Ji−1OWi = IDi Ji and Di is the exceptional divisor of πi. We will say that (6.5) is a resolution of (W, J, b) if Sing(Jn, b) = ∅.

Deﬁnition 6.7. Suppose that r = max νJ , q ∈ Sing(J, r). A non-singular codimension 1 subvariety H of an aﬃne neighborhood U of q in W is called a hypersurface of maximal contact for J at q if 1. Sing(J, r) ∩ U ⊂ H, and 2. if

Wn → · · · → W1 → W is a sequence of monoidal transforms of the form of (6.5) (with b = r), then the strict transform Hn of H on Un = Wn ×W U is such that Sing(Jn, r) ∩ Un ⊂ Hn. 6.2. Hypersurfaces of maximal contact and induction in resolution 67

With the assumptions of Deﬁnition 6.7, a non-singular codimension 1 subvariety H of U = spec(OˆW,q) is called a formal hypersurface of maximal contact for J at q if 1 and 2 of Deﬁnition 6.7 hold, with U = spec(OˆW,q).

If X ⊂ W is a codimension 1 subvariety, with r = max νX , q ∈ max νX , then our definition of hypersurface of maximal contact for J = IX coincides with the definition of a hypersurface of maximal contact for X given in Definition A.20. In this case, the ideal sheaf Jn in (6.5) is the ideal sheaf of the strict transform Xn of X on Wn. Now suppose that X ⊂ W is a singular hypersurface, and K is algebraically closed. Let

r = max{νX } > 1 be the set of points of maximal multiplicity on X. Let q ∈ Sing(IX , r) be a closed point of maximal multiplicity r. The basic strategy of resolution is to construct a sequence

Wn → · · · → W1 → W of monoidal transforms centered at non-singular subvarieties of the locus of points of multiplicity r on the strict transform of X so that we eventually reach a situation where all points of the strict transform of X have multiplicity < r. We know that under such a sequence the multiplicity can never go up, and always remains ≤ r (Lemma 6.4). However, getting the multiplicity to drop to less that r everywhere is much more diﬃcult.

Notice that the desired sequence Wn → W1 will be a resolution of the form of (6.5), with (Ji, b) = (IXi , r), where Xi is the strict transform of X on Wi.

Let q ∈ Sing(IX , r) be a closed point. After Weierstrass preparation and performing a Tschirnhausen transformation, there exist regular parameters (x1, . . . , xn, y) in R = OˆW,q such that there exists f ∈ R such that f = 0 is a local equation of X at q, and r r X r−i (6.6) f = y + ar(x1, . . . , xn)y . i=2

Set H = V (y) ⊂ spec(R). Then H = spec(S), where S = K[[x1, . . . , xn]]. We observe that H is a (formal) hypersurface of maximal contact for X at q. The veriﬁcation is as follows. We will ﬁrst show that H contains a formal neighborhood of the locus of points of maximal multiplicity on X at q. Suppose that Z ⊂ Sing(IX , r) is a subvariety containing q. Let J = IˆZ,q.

Write J = P1 ∩· · ·∩Pr, where the Pi are prime ideals of the same height r in R. Let Ri = RPi . By semi-continuity of νX , we have f ∈ Pi RPi for all 68 6. Resolution of Singularities in Characteristic Zero

∂ ∂r−1f i. Since ∂y : RPi → RPi , we have (r − 1)!y = ∂yr−1 ∈ JRPi for all i. Thus T y ∈ JRPi = J and Z ∩ spec(R) ⊂ H. Now we will verify that if Y ⊂ Sing(IX , r) is a non-singular subvariety, π : W1 → W is the monoidal transform with center Y , X1 is the strict transform of X and q1 is a closed point of X1 such that π(q1) = q which has maximal multiplicity r, then q1 is on the strict transform H1 of H (over the formal neighborhood of q). Suppose that q1 is such a point. That is, −1 ˆ q1 ∈ π (q) ∩ Sing(IX1 , r). We have shown above that y ∈ IY . Without changing the form of (6.6), we may then assume that there is s ≤ n such that x1 = ... = xs = y = 0 are (formal) local equations of Y at q, and there is a formal regular system of parameters x1(1), . . . , xn(1), y(1) in OW1,q1 such that

x1 = x1(1), x = x (1)x (1) for 2 ≤ i ≤ s, (6.7) i 1 i y = x1(1)y(1), xi = xi(1) for s < i ≤ n. i Since Y ⊂ Sing(IX , r), we have qi ∈ (x1, . . . , xs) for all i. Then we have a (formal) local equation f1 = 0 of X1 at q1, where r f X ai f = = y(1)r + y(1)r−i 1 xr xi 1 i=2 1 (6.8) r r X r−i = y(1) + ai(1)(x1(1), . . . , xn(1))y(1) i=2 and y1(1) = 0 is a local equation of H1 at q1, while x1(1) = 0 is a local equation of the exceptional divisor of π. With our assumption that f1 also has multiplicity r, f1 has an expression of the same form as (6.6), so by induction, H is a (formal) hypersurface of maximal contact. We see that (at least over a formal neighborhood of q) we have reduced the problem of resolution of X to some sort of resolution problem on the (formal) hypersurface H. We will now describe this resolution problem. Set r! r! 2 r I = (a2 , . . . , ar ) ⊂ S = K[[x1, . . . , xn]]. Observe that the scheme of points of multiplicity r in the formal neighborhood of q on X coincides with the scheme of points of multiplicity ≥ r! of I, Sing(fR, r) = Sing(I, r!).

However, we may have νR(I) > r!. Let us now consider the eﬀect of the monodial transform π of (6.7) on I. Since H is a hypersurface of maximal contact, π certainly induces 6.2. Hypersurfaces of maximal contact and induction in resolution 69

a morphism H1 → H, where H1 is the strict transform of H. By direct verification, we see that the transform I1 of I (this transform is defined by (6.4)) satisfies 1 I1S1 = r! IS1, x1 ˆ where S1 = K[[x1(1), . . . , xn(1)]] = OH1,q1 , and this ideal is thus, by (6.8), the ideal r! r! I1S1 = (a2(1) 2 , . . . , ar(1) r ), which is obtained from the coefficients of f1, our local equation of X1, in the exact same way that we obtained the ideal I from our local equation f of

X. We further observe that νS1 (I1S1) ≥ r! if and only if νq1 (X1) ≥ r. We see then that to reduce the multiplicity of the strict transform of X over q, we are reduced to resolving a sequence of the form (6.5), but over a formal scheme. The above is exactly the procedure which is followed in the proof of resolution for curves given in Section 3.4, and the proof of resolution for surfaces in Section 5.1. When X is a curve (dim W = 2) the induced resolution problem on the formal curve H is essentially trivial, as the ideal I is in fact just the ideal generated by a monomial, and we only need blow up points to divide out powers of the terms in the monomial. When X is a surface (dim W = 3), we were able to make use of another induction, by realizing that resolution over a general point of a curve in the locus of points of maximal multiplicity reduces to the problem of resolution of curves (over a non-closed field). In this way, we were able to reduce to a resolution problem over finitely many points in the locus of points of maximal multiplicity on X. The potentially worrisome problem of extending a resolution of an object over a formal germ of a hypersurface to a global proper morphism over W was not a great difficulty, since the blowing up of a point on the formal surface H always extends trivially, and the only other kind of blow-ups we had to consider were the blow-ups of non-singular curves contained in the locus of maximal multiplicity. This required a little attention, but we were able to fairly easily reduce to a situation were these formal curves were always algebraic. In this case we first blew up points to reduce to the situation where the ideal I was locally a monomial ideal. Then we blew up more points to make it a principal monomial ideal. At this point, we finished the resolution process by blowing up non-singular curves in the locus of maximal multiplicity on the strict transform of the surface X, which amounts to dividing out powers of the terms in the principal monomial ideal. In Section 5.2, we extended this algorithm to obtain embedded resolution of the surface X. This required keeping track of the exceptional divisors 70 6. Resolution of Singularities in Characteristic Zero which occur under the resolution process, and requiring that the centers of all monoidal transforms make SNCs with the previous exceptional divisors. We introduced the η invariant, which counts the number of components of the exceptional locus which have existed since the multiplicity last dropped. In this chapter we give a proof of resolution in arbitrary dimension (over fields of characteristic zero) which incorporates all of these ideas into a general induction statement. The fact that we must consider some kind of covering by local hypersurfaces which are not related in any obvious way is incorporated in the notion of General Basic Object. In the algorithm of this chapter Tschirnhausen is replaced by a method of Giraud for finding algebraic hypersurfaces of maximal contact for an ideal J of order b, by finding a hypersurface in 4b−1(J).

6.3. Pairs and basic objects

Deﬁnition 6.8. A pair is (W0,E0), where W0 is a non-singular variety over a ﬁeld K of characteristic zero, and E0 = {D1,...,Dr} is an ordered collection of reduced non-singular divisors on W0 such that D1 + ··· + Dr is a reduced simple normal crossings divisor.

Suppose that (W0,E0) is a pair, and Y0 is a non-singular closed subvariety of W0. Y0 is a permissible center for (W0,E0) if Y0 has SNCs with E0. Let π : W1 → W be the monoidal transform of W with center Y0 and exceptional divisor D. By abuse of notation we identify Di (1 ≤ i ≤ r) with its strict transform on W1 (which could become the empty set if Di is a union of components of Y0). By a local analysis, using the regular parameters of the proof of Lemma 6.4, we see that D1 + ··· + Dr + D is a SNC divisor on W1.

Now set E1 = {D1,...,Dr,Dr+1 = D}. We have thus deﬁned a new pair (W1,E1), called the transform of (W0,E0) by the map π. The diagram

(W1,E1) → (W0,E0) will be called a transformation. Observe that some of the strict transforms Di may be the empty set.

Deﬁnition 6.9. Suppose that (W0,E0) is a pair with E0 = {D1,...,Dr}, and W˜ 0 is a non-singular subvariety of W0. Deﬁne scheme-theoretic intersec- tions D˜i = Di · W˜ 0 for all i. Suppose that D˜i are non-singular for 1 ≤ i ≤ r and that D˜1 + ··· + D˜r is a SNC divisor. Set E˜0 = {D˜1,..., D˜r}. Then (W˜ 0, E˜0) is a pair, and

(W˜ 0, E˜0) → (W0,E0) is called an immersion of pairs. 6.3. Pairs and basic objects 71

Deﬁnition 6.10. A basic object is (W0, (J0, b),E0), where (W0,E0) is a pair, J0 is an ideal sheaf of W0 which is non-zero on all connected components of W0, and b is a natural number. The singular locus of (W0, (J0, b),E0) is b−1 Sing(J0, b) = {q ∈ W0 | νq(J0) ≥ b} = V (4 (J0)) ⊂ W0.

Deﬁnition 6.11. A basic object (W0, (J0, b),E0) is a simple basic object if

νq(J) = b if q ∈ Sing(J, b).

Suppose that (W0, (J0, b),E0) is a basic object and Y0 is a non-singular closed subvariety of W0. Y0 is a permissible center for (W0, (J0, b),E0) if Y0 is permissible for (W0,E0) and Y0 ⊂ Sing(J0, b). Suppose that Y0 is a permissible center for (W0, (J0, b),E0). Let π : W1 → W0 be the monoidal transform of W0 with center Y0 and exceptional divisor D. Thus we have a transformation of pairs (W1,E1) → (W0,E0). Let J 1 be the weak transform of J0 on W1, so that there is a locally constant function c1 on D (constant on each connected component of D) such that the total transform of J0 is

c1 J0OW1 = ID J 1.

We necessarily have c1 ≥ b, so we can deﬁne

−b c1−b J1 = ID J0OW1 = ID J 1.

We have thus deﬁned a new basic object (W1, (J1, b),E1), called the transform of (W0, (J0, b),E0) by the map π. The diagram

(W1, (J1, b),E1) → (W0, (J0, b),E0) will be called a transformation. Observe that a transform of a basic object is a basic object, and (by Lemma 6.4) the transform of a simple basic object is a simple basic object. Suppose that

(6.9) (Wk, (Jk, b),Ek) → · · · → (W0, (J0, b),E0) is a sequence of transformations.

Definition 6.12. (6.9) is a resolution of (W0, (J0, b),E0) if Sing(Jk, b) = ∅. Definition 6.13. Suppose that (6.9) is a sequence of transformations. Set J 0 = J0, and let J i+1 be the weak transform of J i for 0 ≤ i ≤ k − 1. For 0 ≤ i ≤ k define 1 w-ord : Sing(J , b) → i i b Z by ν (J ) w-ord (q) = q i i b 72 6. Resolution of Singularities in Characteristic Zero and define 1 ord : Sing(J , b) → i i b Z by ν (J ) ord (q) = q i . i b

If we assume that Yi ⊂ Max w-ordi ⊂ Sing(Ji, b) for 1 ≤ i ≤ k in (6.9), then max w-ord0 ≥ · · · ≥ max w-ordk by Lemma 6.4. Definition 6.14. Suppose that (6.9) is a sequence of transformations such that Yi ⊂ Max w-ordi ⊂ Sing(Ji, b) for 0 ≤ i < k. Let k0 be the smallest index such that max w-ordk0−1 > max w-ordk0 = ··· = max w-ordk. In particular, k0 is defined to be 0 if max w-ord0 = ··· = max w-ordk. Set + − − Ek = Ek ∪ Ek , where Ek is the ordered set of divisors D in Ek which are + − strict transforms of divisors of Ek0 and Ek = Ek − Ek . Define tk : Sing(Jk, b) → Q × Z, where Q × Z has the lexicographic order by tk(q) = (w-ordk(q), η(q)), Card{D ∈ Ek | q ∈ D} if w-ordk(q) < max w-ordk0 , η(q) = − Card{D ∈ Ek | q ∈ D} if w-ordk(q) = max w-ordk0 . This definition allows us to inductively define upper semi-continuous functions t0, t1, . . . , tk on a sequence of transformations (6.9) such that Yi ⊂ Max w-ordi ⊂ Sing(Ji, b) for 1 ≤ i ≤ k. Lemma 6.15. Suppose that (6.9) is a sequence of transformations such that Yi ⊂ Max ti ⊂ Max w-ordi for 0 ≤ i ≤ k. Then

max tk ≤ · · · ≤ max tk. Lemma 6.15 follows from Lemma 6.4 and a local analysis, using the regular parameters of the proof of Lemma 6.4.

Deﬁnition 6.16. Suppose that (W0, (J0, b),E0) is a basic object, and

(6.10) (Wk, (Jk, b),Ek) → · · · → (W0, (J0, b),E0) is a sequence of transformations such that max w-ordk = 0. Then we say that (Wk, (Jk, b),Ek) is a monomial basic object (relative to (6.10)).

Suppose that (Wk, (Jk, b),Ek) is a monomial basic object (relative to a sequence (6.10)). Let E0 = {D1,...,Dr} and let Di be the exceptional divisor of πi : Wi → Wi−1, where i = i + r. Then we have an expansion (in a neighborhood of Sing(Jk, b)) of the weak transform J k of J:

d1 dk J k = I ···I , D1 Dk 6.3. Pairs and basic objects 73

where the di are functions on Wk which are zero on Wk − Di and di is a locally constant function on Di for 1 ≤ i ≤ k.

Suppose that B = (Wk, (Jk, b),Ek) is a monomial basic object (relative to (6.10)). We deﬁne

Γ(B) : Sing(Jk, b) → IM = Z × Q × ZN by Γ(B)(q) = (−Γ1(q), Γ2(q), Γ3(q)), where    p such that there exist i1, . . . , ip,  Γ1(q) = min with r < i1, . . . , ip ≤ r + k, such that ,   di1 (q) + ··· + dip (q) ≥ b for some q ∈ Di1 ∩ · · · ∩ Dip   di1 (q)+···+dip (q)  b , with r < i1, . . . , ip ≤ r + k,  Γ2(q) = max such that p = Γ1(q), ,    di1 (q) + ··· + dip (q) ≥ b, q ∈ Di1 ∩ · · · ∩ Dip  ( ) di1 (q)+···+dip (q) (i1, . . . , ip, 0,...) such that Γ2(q) = , Γ3(q) = max b . with r < i1, . . . , ip ≤ r + k, q ∈ Di1 ∩ · · · ∩ Dip Observe that Γ(B) is upper semi-continuous, and Max Γ(B) is non-singular.

Lemma 6.17. Suppose that B = (Wk, (Jk, b),Ek) is a monomial basic object (relative to a sequence (6.10)). Then Yk = Max Γ(B) is a permissible center for B, and if (Wk+1, (Jk+1, b),Ek+1) → (Wk, (Jk, b),Ek) is the transformation induced by the monoidal transform πk+1 : Wk+1 → Wk with center Yk, then B1 = (Wk+1, (Jk+1, b),Ek+1) is a monomial basic object and

max Γ(B) > max Γ(B1) in the lexicographic order.

Proof. Max Γ(B) is certainly a permissible center. We will show that

max Γ(B) > max Γ(B1).

We will ﬁrst assume that max(−Γ1) < −1. Let i = i + r. There exist locally constant functions di on W such that di(q) = 0 if q 6∈ Di and di is locally constant on Di such that

d1 dk Jk = I ···I . D1 Dk

Let Di be the strict transform of Di on Wk+1 and let Dk+1 be the exceptional divisor of the blow-up πk+1 : Wk+1 → Wk of Yk = Max Γ(B). Then d d1 k+1 Jk+1 = I ···I , D1 Dk+1 74 6. Resolution of Singularities in Characteristic Zero

where for 1 ≤ i ≤ k we have di(q) = di(πk+1(q)) if q ∈ Di and di(q) = 0 if q 6∈ Di. We have

dk+1(q) = di1 (πk+1(q)) + ··· + dip (πk+1(q)) − b if q ∈ Dk+1, and dk+1(q) = 0 if q 6∈ Dk+1. For q ∈ Sing(Jk+1, b) we have Γ(B1)(q) = Γ(B)(πk+1(q)) if πk+1(q) 6∈ Max Γ(B), so it suﬃces to prove that

Γ(B1)(q) < Γ(B)(πk+1(q)) if q ∈ Sing(Jk+1, b) and πk+1(q) ∈ Max Γ(B). Suppose that

Γ(B1)(q) = (−p1, w1, (j1, . . . , jp1 , 0,...)) and Γ(B)(πk+1(q)) = max Γ(B) = (−p, w, (i1, . . . , ip, 0,...)). We will ﬁrst verify that

(6.11) p1 = Γ1(q) ≥ Γ1(πk+1(q)) = p.

If k + 1 6∈ {j1, . . . , jp1 }, then the inequality b ≤ d (q) + ··· + d (q) ≤ d (π (q)) + ··· + d (π (q)) j1 jp1 j1 k+1 jp1 k+1 implies p1 ≥ p.

Suppose that k + 1 ∈ {j1, . . . , jp1 }, so that j1 = k + 1. If p1 < p, there exists ik ∈ {i1, . . . , ip} such that ik 6∈ {j1, . . . , jp1 }. Let d = d (q) + ··· + d (q). j1 jp1 Then d = d (q) + ··· + d (q) + d (π (q)) + ··· + d (π (q)) − b j2 jp1 i1 k+1 ip k+1

≤ (di1 (πk+1(q)) + ··· + dik−1 (πk+1(q)) + dik+1 (πk+1(q))

+ ··· + dip (πk+1(q)) − b) +d (π (q)) + d (π (q)) + ··· + d (π (q)) ik k+1 j2 k+1 jp1 k+1 < d (π (q)) + d (π (q)) + ··· + d (π (q)) ik k+1 j2 k+1 jp1 k+1 < b, a contradiction. Thus p1 ≥ p.

Now assume that p1 = p. We will verify that w1 ≤ w. If k + 1 6∈

{j1, . . . , jp1 }, we have d (q)+···+d (q) j1 jp1 w1 = b d (π (q))+···+d (π (q)) j1 k+1 jp1 k+1 ≤ b ≤ w.

If k + 1 ∈ {j1, . . . , jp1 }, then j1 = k + 1. We must have d (π (q)) + ··· + d (π (q)) < b, j2 k+1 jp1 k+1 6.4. Basic objects and hypersurfaces of maximal contact 75 so that w b = d (π (q)) + ··· + d (π (q)) − b + d (q) + ··· + d (q) 1 i1 k+1 ip k+1 j2 jp1

≤ di1 (πk+1(q)) + ··· + dip (πk+1(q)) + d (π (q)) + ··· + d (π (q)) − b j2 k+1 jp1 k+1 < wb.

Thus w1 < w.

Assume that p = p1 and w = w1. We have shown that k + 1 6∈

{j1, . . . , jp1 } in this case. Since

Di1 ∩ · · · ∩ Dip = ∅, we must have (j1, . . . , jp, 0,...) < (i1, . . . , ip, 0,...).

It remains to verify the lemma in the case when max(−Γ1(B)) = −1. This case is not diﬃcult, and is left to the reader. 6.4. Basic objects and hypersurfaces of maximal contact

λ Suppose that B = (W, (J, b),E) is a basic object and {W }λ∈Λ is a (ﬁnite) open cover of W . We then have associated basic objects Bλ = (W λ, (J λ, b),Eλ) for λ ∈ Λ obtained by restriction to W λ. Observe that Sing(J λ, b) = Sing(J, b) ∩ W λ.

If B1 = (W1, (J1, b),E1) → (W, (J, b),E) is a transformation induced by a monoidal transform π : W1 → W , then there are associated transformations λ λ λ λ λ λ λ B1 = (W1 , (J1 , b),E1 ) → (W , (J , b),E ), λ −1 λ λ λ λ where W1 = π (W ) and J1 = J1 | W1 . In particular, {W1 } is an open cover of W1. 1 Deﬁnition 6.18. Let W1 = W × AK with projection π : W1 → W . Sup- pose that (W, (J, b),E) is a basic object. Then there is a basic object 0 −1 (W1, (J1, b),E1), where J1 = JOW1 . If E = {D1,...,Dr} and Di = π (Di) 0 0 for 1 ≤ i ≤ r, then E1 = {D1,...,Dr}. The diagram

(W1, (J1, b),E1) → (W, (J, b),E) is called a restriction.

Remark 6.19. Suppose that (W, (J, b),E) is a basic object and Wh ⊂ b−1 W is a non-singular hypersurface. Suppose that IWh ⊂ 4 (J). Then b−1 b−1 Sing(J, b) ⊂ V (4 (J)) ⊂ Wh, and νq(4 (J) = 1 for any q ∈ Sing(J, b). Conversely, if (W, (J, b),E) is a simple basic object and q ∈ Sing(J, b), b−1 then νq(4 (J)) = 1, so there exist an aﬃne neighborhood U of q in W and a non-singular hypersurface Uh ⊂ U such that U ∩ Sing(J, b) ⊂ Uh. 76 6. Resolution of Singularities in Characteristic Zero

Lemma 6.20. Suppose that

(W1, (J1, b),E1) → (W, (J, b),E) is a transformation of basic objects. Let D be the exceptional divisor of W1 → W . Then for all integers i with 1 ≤ i ≤ b we have b−i i 4W (J)OW1 ⊂ ID and 1 4b−i(J)O ⊂ 4b−i(J ). i W W1 W1 1 ID

Proof. Let Y ⊂ Sing(J, b) be the center of π : W1 → W . b−i i The first inclusion follows since νq(4W (J)) ≥ i if q ∈ Y , so that ID b−i divides 4W (J)OW1 . Now we will prove the second inclusion. The second inclusion is trivial if b i = b, since JOW1 = IDJ1 by definition. We will prove the second inclusion by descending induction on i. Suppose that the inclusion is true for some i > 1. Let q ∈ D be a closed point, p = π(q). Let K0 = K(q). By Lemma 0 A.16, it suffices to prove the inclusion on W ×K K , so we may assume that K(q) = K. Then there exist regular parameters x0, x1, . . . , xn in R = OW,p such that I = (x , x , . . . , x ), with m ≤ n; x , x1 ,..., xm , x , . . . , x , Y,p 0 1 m 0 x0 x0 m+1 n is a regular system of parameters in R1 = OW1,q, and ID,q = (x0). It suffices b−(i−1) to show that for f ∈ 4W (J)p, f (6.12) ∈ 4b−(i−1)(J ) . i−1 W1 1 q x0 b−i b−(i−1) If f ∈ 4W (J)p ⊂ 4W (J)p, then the formula follows by induction, so b−i we may assume that f = D(g) with g ∈ 4W (J)p, D an R-derivation. Set D0 = x D. D0 is an R -derivation on R = O since D0( xi ) ∈ O for 0 1 1 W1,q x0 W1,q 1 ≤ i ≤ m. By induction, g 1 ∈ 4b−i(J)O ⊂ 4b−i(J ) ⊂ 4b−(i−1)(J ) . i i W W1,q W1 1 q W1 1 q x0 ID Thus g D0( ) ∈ 4b−(i−1)(J ) , i W1 1 q x0 0 g D(g) g D ( i ) = i−1 − iD(x0) i , x0 x0 x0 which implies f D(g) g g = = D0( ) + iD(x ) ∈ 4b−(i−1)(J ) , i−1 i−1 i 0 i W1 1 q x0 x0 x0 x0 and (6.12) follows. 6.4. Basic objects and hypersurfaces of maximal contact 77

Lemma 6.21. Suppose that (W, (J, b),E) is a simple basic object, and Wh ⊂ b−1 W is a non-singular hypersurface such that IWh ⊂ 4 (J). Suppose that

(W1, (J1, b),E1) → (W, (J, b),E) is a transformation with center Y . Let (Wh)1 be the strict transform of Wh on W1. Then Y ⊂ Wh, (Wh)1 is a non-singular hypersurface in W1, and b−1 I(Wh)1 ⊂ 4 (J1).

Proof. Let D be the exceptional divisor of π : W1 → W . Suppose that q ∈ D is a closed point, and b−1 p = π(q) ∈ Y ⊂ Sing(J, b) = V (4 (J)) ⊂ Wh. Recall that Y ⊂ Sing(J, b) by the definition of transformation. Let f = 0 be a local equation for Wh at p, and g = 0 a local equation of D at q. Then f f b−1 g = 0 is a local equation of (Wh)1 at q, and g ∈ 4 (J1)q by Lemma 6.20. Thus f Sing(J , b) = V (4b−1(J )) ⊂ V ( ). 1 1 g Definition 6.22. Suppose that B = (W, (J, b),E) is a simple basic object ˜ b−1 and W is a non-singular closed subvariety of W such that IW˜ ⊂ 4 (J). Then the coefficient ideal of B on W˜ is b−1 b! X i b−i C(J) = 4 (J) OW˜ . i=0 Lemma 6.23. With the notation of Lemma 6.21 and Definition 6.22,

Sing(J, b) = Sing(C(J), b!) ⊂ Wh ⊂ W.

Proof. It suﬃces to check this at closed points q ∈ Wh. By Lemma A.16 we may assume that K(q) = K. Then there exist regular parameters x1, . . . , xn at q in W such that x1 = 0 is a local equation for Wh at q. Let R = OˆW,q = ˆ K[[x1, . . . , xn]], S = OWh,q = K[[x2, . . . , xn]] with natural projection R → S. It suﬃces to show that ˆ j (6.13) νR(Jˆ) ≥ b if and only if νS(4 (JSˆ )) ≥ b − j for j = 0, 1, . . . , b − 1. Suppose that f ∈ Jˆ ⊂ R. Write

X i f = aix1 i≥0 with ai ∈ S for all i. We have νR(f) ≥ b if and only if νS(aj) ≥ b − j for j = 0, 1, . . . , b − 1. 78 6. Resolution of Singularities in Characteristic Zero

ˆ j ˆ 1 ∂j f We also have aj ∈ 4 (J)S, since aj is the projection in S of j! j . Thus ∂x1 we have the if direction of (6.13). The ideal 4ˆ j(Jˆ) is spanned by elements of the form

∂i1+···+in f ci ,...,i = 1 n i1 in ∂x1 ··· ∂xn ˆ with f ∈ J and i1 + ··· + in ≤ j, and the projection of ci1,...,in in S is

i2+···+in ∂ ai1 i1! . i2 in ∂x2 ··· ∂xn Thus we have the only if direction of (6.13). Theorem 6.24. Suppose that (W, (J, b),E) is a simple basic object, and λ {W }λ∈Λ is an open cover of W such that for each λ there is a non-singular λ λ b−1 λ λ hypersurface W ⊂ W such that I λ ⊂ 4 (J ) and W has simple h Wh h λ λ normal crossings with E . Let Eh be the set of non-singular reduced divisors λ λ on Wh obtained by intersection of E with Wh . Recall that R(1)(Z) denotes the codimension 1 part of a closed reduced subscheme Z, as deﬁned in (6.1). 1. If R = R(1)(Sing(J, b)) 6= ∅, then R is non-singular, open and closed in Sing(J, b), and has simple normal crossings with E. Fur- thermore, the monoidal transform of W with center R induces a transformation

(W1, (J1, b),E1) → (W, (J, b),E)

such that R(1)(Sing(J1, b)) = ∅. 2. If R(1)(Sing(J, b)) = ∅, for each λ ∈ Λ, the basic object λ λ λ (Wh , (C(J ), b!),Eh ) has the following properties: a. Sing(J λ, b) = Sing(C(J λ), b!). b. Suppose that

(6.14) (Ws, (Js, b),Es) → · · · → (W, (J, b),E) is a sequence of transformations and restrictions, with induced sequences for each λ λ λ λ λ λ λ (Ws , (Js , b),Es ) → · · · → (W , (J , b),E ). Then for each λ there is an induced sequence of transformations and restrictions λ λ λ λ λ λ ((Wh)s , (C(J )s, b!), (Eh)s ) → (Wh , (C(J ), b!),Eh ) 6.4. Basic objects and hypersurfaces of maximal contact 79

and a diagram where the vertical arrows are closed immersions of pairs λ λ λ λ (Ws ,Es ) → · · · → (W ,E ) (6.15) ↑ ↑ λ λ λ λ ((Wh)s , (Eh)s ) → · · · → (Wh ,Eh ) such that λ λ Sing(C(J )i, b!) = Sing(Ji , b) for all i.

Proof. We may assume that W λ = W .

First suppose that T = R(1)(Sing(J, b)) 6= ∅. Then T ⊂ Sing(J, b) ⊂ Wh implies T is a union of connected components of Wh, so T is non-singular, open and closed in Sing(J, b), and has SNCs with E. Suppose that q ∈ T . Let f = 0 be a local equation of T at q in W . Since b (W, (J, b),E) is a simple basic object, Jq = f OW,q. If π1 : W1 → W is the monoidal transform with center T , then (J1)q1 = OW,q1 for all q1 ∈ W1 such that π1(q1) ∈ T , and necessarily, R(1)(Sing(J1, b)) = ∅. Now suppose that R(1)(Sing(J, b)) = ∅, and (6.14) is a sequence of transformations and restrictions of simple basic objects.

Suppose that (W1, (J1, b),E1) → (W, (J, b),E) is a transformation with center Y0. By Lemma 6.21 we have Y0 ⊂ Wh, the strict transform (Wh)1 of Wh is a non-singular hypersurface in W1, and b−1 I(Wh)1 ⊂ 4 (J1).

Since Wh has SNCs with E, Y ⊂ Wh and Y has SNCs with E, it follows that Y has SNCs with E ∪ Wh. Thus (Wh)1 has SNCs with E1, and if (Eh)1 = E1 · (Wh)1 is the scheme-theoretic intersection, we have a closed immersion of pairs

((Wh)1, (Eh)1) → (W1,E1).

Since the case of a restriction is trivial, and each (Wi, (Ji, b),Ei) is a simple basic object, we thus can conclude by induction that we have a diagram of closed immersions of pairs (6.15) such that b−1 (6.16) I(Wh)j ⊂ 4 (Jj) for all j. We may assume that the sequence (6.14) consists only of transformations. Let Dk be the exceptional divisor of Wk → Wk−1. It remains to show that

(6.17) Sing(C(J)j, b!) = Sing(Jj, b) 80 6. Resolution of Singularities in Characteristic Zero for all j. By Lemma A.16 and Remark A.18 we may assume that K is algebraically closed. For k > 0 and 0 ≤ i ≤ b − 1, set

h b−i i 1 h b−i i 4 (J) = 4 (J) OW , k Ii k−1 k Dk so that b b! X h b−i i i C(J)k = 4 (J) O(W ) . k h k i=1 We will establish the following formulas (6.18) and (6.19) for 0 ≤ k ≤ s and 0 ≤ i ≤ b − 1: h b−i i b−i (6.18) 4 (J) ⊂ 4 (Jk). k ˆ Suppose that q ∈ Sing(C(J0)k, b!) is a closed point. Set Rk = OWk,q, Sk = ˆ O(Wh)k,q. Then there are regular parameters (xk,1, . . . , xk,n, zk) in Rk such that I(Wh)k,q = (zk), and there are generators {fk} of JkRk such that X α (6.19) fk = ak,αzk α with ak,α ∈ K[[xk1, . . . , xk,n]] = Sk such that for 0 ≤ α ≤ b − 1 b! (ak,α) b−α ∈ C(J)kSk.

Assume that the formulas (6.18) and (6.19) hold. (6.18) implies C(J)k ⊂ C(Jk), so by Lemma 6.23, Sing(Jk, b) ⊂ Sing(C(J)k, b!). Now (6.19) shows that Sing(C(J)k, b!) ⊂ Sing(Jk, b), so that Sing(C(J)k, b!) = Sing(Jk, b), as desired. We will now verify formulas (6.18) and (6.19). (6.18) is trivial if k = 0, and (6.19) with k = 0 follows from the proof of Lemma 6.23. We now assume that (6.18) and (6.19) are true for k = t, and prove them for k = t + 1. Since h b−i i b−i 4 (J) ⊂ 4 (Jt), t we have h b−i i 1 h b−i i 1 b−i b−i 4 (J) = 4 (J) ⊂ 4 (Jt) ⊂ 4 (Jt+1), t+1 (I )i t (I )i Dt+1 Dt+1 where the last inclusion follows from the second formula of Lemma 6.20. We have thus veriﬁed formula (6.18) for k = t + 1.

Let qt+1 ∈ Sing(C(J0)t+1, b!) be a closed point, and qt the image of qt+1 in {Sing(C(J0)t, b!).

By assumption, there exist regular parameters (xt1, . . . , xtn, zt) in Rt which satisfy (6.19). Recall that we have reduced to the assumption that 6.5. General basic objects 81

K is algebraically closed. Let Yt be the center of Wt+1 → Wt. By (6.16), there exist regular parameters (x1,..., xn) in St = K[[xt1, . . . , xtn]] ⊂ Rt such that (x1,..., xn, zt) are regular parameters in Rt, there exists r with ˆ r ≤ n such that x1 = ··· = xr = zt = 0 are local equations of Yt in OWt,qt , ˆ and Rt+1 = OWt+1,qt+1 has regular parameters (xt+1,1, . . . , xt+1,n, zt+1) such that x1 = xt+1,1, xi = xt+1,1xt+1,i for 2 ≤ i ≤ r, zt = xt+1,1zt+1, xi = xt+1,i for r < i ≤ n.

In particular, xt+1,1 = 0 is a local equation of the exceptional divisor Dt+1 and zt+1 = 0 is a local equation of (Wh)t+1. We have generators

ft X α ft+1 = b = at+1,αzt+1 xt+1,1 α 1 of Jt+1Rt+1 = b JtRt+1 such that xt+1,1

at,α at+1,α = b−α xt,1 if α < b, and

b! 1 b! 1 (a ) b−α = (a ) b−α ∈ C(J) S = C(J) S . t+1,α (x )b! t,α (I )b! t t+1 t+1 t+1 t,1 Dt+1

6.5. General basic objects

Deﬁnition 6.25. Suppose that (W0,E0) is a pair. We deﬁne a general basic object (GBO) (F0,W0,E0) on (W0,E0) to be a collection of pairs (Wi,Ei) with closed sets Fi ⊂ Wi which have been constructed inductively to satisfy the following three properties:

1. F0 ⊂ W0.

2. Suppose that Fi ⊂ Wi has been deﬁned for the pair (Wi,Ei). If πi+1 :(Wi+1,Ei+1) → (Wi,Ei) is a restriction, then Fi+1 = −1 πi+1(Fi). 3. Suppose that Fi ⊂ Wi has been deﬁned for the pair (Wi,Ei). If πi+1 :(Wi+1,Ei+1) → (Wi,Ei) is a transformation centered at ∼ Yi ⊂ Fi, then Fi+1 ⊂ Wi+1 is a closed set such that Fi − Yi = Fi+1 − πi+1(Yi) by the map πi+1.

The closed sets Fi will be called the closed sets associated to (F0,W0,E0). 82 6. Resolution of Singularities in Characteristic Zero

(W1,E1) → (W0,E0) is a transformation with center Y0 ⊂ F0, then we can use the pairs (Wi,Ei) and closed subsets Fi ⊂ Wi in Deﬁnition 6.25 which map to (W1,E1) by a sequence of restrictions and transformations satisfying 2 and 3 to deﬁne a general basic object (F1,W1,E1) over (W1,E1). We will say that Y0 is a permissible center for (F0,W0,E0) and call

(F1,W1,E1) → (F0,W0,E0) a transformation of general basic objects. If

(W1,E1) → (W0,E0) is a restriction, then we can use the pairs (Wi,Ei) and closed subsets Fi defined in Definition 6.25 above which map to (W1,E1) by a sequence of restrictions and transformations satisfying (2) and (3) to define a general basic object (F1,W1,E1) over (W1,E1). We will call

(F1,W1,E1) → (F0,W0,E0) a restriction of general basic objects. A composition of restrictions and transformations of general basic objects

(6.20) (Fr,Wr,Er) → · · · → (F0,W0,E0) will be called a permissible sequence for the GBO (F0,W0,E0). (6.20) will be called a resolution if Fr = ∅.

Deﬁnition 6.26. Suppose that (W0,E0) is a pair, and (F0,W0,E0) is a GBO over (W0,E0), with associated closed sets {Fi}. Further, suppose that λ {W0 }λ∈Λ is a ﬁnite open covering of W0 and d ∈ N. λ Suppose that we have a collection of basic objects {B0 }λ∈Λ with λ ˜ λ λ λ ˜λ B0 = (W0 , (a0 , b ), E0 ). λ We say that {B0 }λ∈Λ is a d-dimensional structure on the GBO (F0,W0,E0) if for each λ ∈ Λ 1. we have an immersion of pairs λ ˜ λ ˜λ λ λ j0 :(W0 , E0 ) → (W0 ,E0 ), ˜ λ where W0 has dimension d, and 2. if

(Fr,Wr,Er) → · · · → (F0,W0,E0) 6.6. Functions on a general basic object 83

is a permissible sequence for the GBO (F0,W0,E0) (with associated closed sets Fi ⊂ Wi for 0 ≤ i ≤ r), then the sequence of morphisms of pairs λ λ λ λ (Wr ,Er ) → · · · → (W0 ,E0 ) induces diagrams of immersions of pairs λ λ λ λ (Wr ,Er ) → · · · → (W0 ,E0 ) ↑ ↑ ˜ λ ˜λ ˜ λ ˜λ (Wr , Er ) → · · · → (W0 , E0 ) such that ˜ λ λ λ ˜λ ˜ λ λ λ ˜λ (Wr , (ar , b ), Er ) → · · · → (W0 , (a0 , b ), E0 ) is a sequence of transformations and restrictions of basic objects, and λ λ λ λ Fi = Fi ∩ Wi = Sing(ai , b ) for 0 ≤ i ≤ r.

6.6. Functions on a general basic object

Proposition 6.27. Let (F0,W0,E0) be a GBO which admits a d-dimensional structure, and fix notation as in Definition 6.26. Let λ λ λ F0 = Sing(a0 , b ). d There is a function ord : F0 → Q such that for all λ ∈ Λ, the composition λ d λ j0 ord F0 → F0 → Q λ λ is the function ord : F0 → Q defined by Definition 6.13 for λ λ λ F0 = Sing(a0 , b ). λ λ ˜ λ λ λ ˜λ Proof. Let {B0 }λ∈Λ, where B0 = (W0 , (a0 , b ), E0 ), be the given d-dimensional structure on (F0,W0,E0). Let E0 = {D1,...,Dr}. We have closed λ ˜ λ λ immersions j0 : W0 → W0 for all λ ∈ Λ. Suppose that q0 ∈ F0 is a closed point, and λ, β ∈ Λ are such that there λ ˜ λ β ˜ β λ λ β β are q0 ∈ W0 and q0 ∈ W0 such that j0 (q0 ) = j0 (q0 ) = q0. We must show that λ β νqλ (a0 ) νqβ (a0 ) 0 = 0 . bλ bβ We will prove this by expressing these numbers intrinsically, in terms of the GBO (F0,W0,E0), so that they are independent of λ. We will carry out the λ calculation for B0 . π1 1 Let (F1,W1,E1) → (F0,W0,E0) be the restriction where W1 = W0×AK . 1 −1 Let q1 = (q0, 0) ∈ W1. Let L1 = q0 × AK ; then L1 ⊂ F1 = π1 (F0). 84 6. Resolution of Singularities in Characteristic Zero

For any integer N > 0 we can construct a permissible sequence of pairs

πN π2 π1 (6.21) (WN ,EN ) → · · · → (W1,E1) → (W0,E0) where for i ≥ 1, Wi+1 → Wi is the blow-up of a point qi. qi is deﬁned as follows. Let Li be the strict transform of L1 on Wi, and let Di be the exceptional divisor of πi, where i = i + r. Then qi = Li ∩ Di.

We have q1 ∈ L1 ⊂ F1. We must have L2 ⊂ F2 by 3 of Deﬁnition 6.25. Thus π2 induces a transformation of GBOs (F2,W2,E2) → (F1,W1,E1), and q2 ∈ F2. By induction, we have for any N > 0 a permissible sequence

πN π2 π1 (6.22) (FN ,WN ,EN ) → · · · → (F1,W1,E1) → (F0,W0,E0).

λ λ (6.22) induces a permissible sequence of basic objects with q1 = (q0 , 0) λ and centers qi for i ≥ 1: ˜ λ λ λ ˜λ πN π1 ˜ λ λ λ ˜λ (6.23) (WN , (aN , b ), EN ) → · · · → (W0 , (a0 , b ), E0 ). ˜ λ λ ˜ λ Observe that Li ⊂ Wi for all i ≥ 1, so that qi ∈ Wi for all i ≥ 1. Set 0 λ b = ν λ (a ). q0 0 λ ˜ λ Suppose that (x1, . . . , xd) are regular parameters at q0 in W0 . Then λ ˜ λ 1 (x1, . . . , xd, t) are regular parameters at q1 in W1 (where AK = spec(K[t])), ˜ λ and thus there are regular parameters (x1(N), x2(N), . . . , xd(N), t) in WN λ N−1 at qN which are deﬁned by xi = xi(N)t for 1 ≤ i ≤ d. t = 0 is a local equation of the exceptional divisor D˜ λ of W˜ λ → W˜ λ . N N N−1 0 λ Suppose that f ∈ O ˜ λ λ and ν λ (f) = r. Let K = K(q ). We have an W0 ,q0 q0 0 expression

X i1 id 0 ˆ f = ai1,...,id x ··· x ∈ K [[x1, . . . , xd]] = O ˜ λ λ , 1 d W0 ,q0 i1+···id≥r 0 0 ˆ where ai1,...,id ∈ K . Thus in K [[x1(N), . . . , xd(N), t]] = O ˜ λ λ we have WN ,qN an expansion

X i1 id (i1+···+id)(N−1) f = ai1,...,id x1(N) ··· xd(N) t i +···+i ≥r 1 d  

(N−1)r X i1 id N−1 = t  ai1,...,id x1(N) ··· xd(N) + t ΩN  . i1+···+id=r

(N−1)r Thus we have an expression f = t fN in O ˜ λ , where t fN . WN ,qN - λ 0 Since ν λ (a ) = b , we have q0 0

λ (N−1)b0 λ a0 O ˜ λ = I λ KN , WN D˜ N 6.6. Functions on a general basic object 85

λ where ID˜ λ - KN . We thus have an expression N 0 λ (6.24) aλ = I(N−1)(b −b )Kλ , N D˜ λ N N λ λ λ ˜ λ λ λ and ν λ (a ) ≥ b since q ∈ FN ∩ W = Sing(a , b ). qN N N N N Under the closed immersion W˜ λ ⊂ W λ we have (for N ≥ 1) D˜ λ = N N N D · W˜ λ , where D˜ λ has dimension d and is irreducible. Since F λ ⊂ W˜ λ , N N N N N it follows that λ ˜ λ dim(FN · DN ) = dim(FN · DN ) ≤ dim DN = d, and the following three conditions are equivalent: λ 1. dim(FN · DN ) = dim(FN · DN ) = d. ˜ λ λ 2. DN ⊂ FN . 3. (N − 1)(b0 − bλ) ≥ bλ. b0 0 λ Thus bλ = 1 holds if and only if b − b = 0, which holds if and only if, for any N, dim(FN · DN ) < d. Thus the condition λ 0 νqλ (a0 ) b 0 = = 1 bλ bλ is independent of λ. Now assume that b0 − bλ > 0. Then, for N suﬃcently large, (N − 1)(b − bλ) ≥ bλ, and F · D = D˜ λ is a permissible center of dimension d for (F ,W ,E ). N N N N n N We now deﬁne an extension of (6.22) by a sequence of transformations (6.25) πN+S πN+1 πN π1 (FN+S,WN+S,EN+S) → · · · → (FN ,WN ,EN ) → · · · → (F0,W0,E0), where πN+i is the transformation with center YN+i = FN+i · DN+i, and we assume that dim YN+i = d for i = 0, 1,...,S − 1. Observe that ˜ λ ˜ λ (6.26) dim YN+i = d if and only if DN+i = DN+i · WN+i ⊂ FN+i. We thus have Y = D˜ λ . (6.26) induces an extension of (6.23) by a N+i N+i sequence of transformations π (W˜ λ , (aλ , bλ), E˜ ) N→+S · · · (6.27) N+S N+S N+S πN+1 ˜ λ λ λ ˜ πN π1 ˜ λ λ λ ˜ → (WN , (aN , b ), EN ) → · · · → (W0 , (a0 , b ), E0). The centers of π in (6.26) are D˜ λ for j = N,...,N +S −1. Thus W˜ → i j N+i+1 W˜ N+i is the identity for i = 0,...,S − 1. 86 6. Resolution of Singularities in Characteristic Zero

We see then that 0 λ λ aλ = I(N−1)(b −b )−ib Kλ N+i D˜ λ N N+i for i = 1,...,S. The statement that the extension (6.27) is permissible is precisely the statment that (N − 1)(b0 − bλ) ≥ Sbλ, which is equivalent to the statement that dim FN+i · DN+i = d for i = 0, 1,...,S − 1. This last statement is equivalent to (N − 1)(b0 − bλ) S ≤ ` = b c, N bλ where bxc denotes the greatest integer in x. Thus `N is determined both by N and by the GBO (F0,W0,E0). The limit ` b0 lim N = − 1 N→∞ N − 1 bλ is deﬁned in terms of the GBO, hence is independent of λ.

Theorem 6.28. Let (F0,W0,E0) be a GBO which admits a d-dimensional structure, and fix notation as in Definition 6.26. Let Fi be the closed sets λ λ λ λ λ associated to the GBO. Let Fi = Sing(ai , b ), and let w-ordi , ti be the λ functions of Definition 6.13 defined for Fi . Consider any permissible sequence of transformations

πr π1 (Fr,Wr,Er) → · · · → (F0,W0,E0). d 1. Then for 1 ≤ i ≤ r, there are functions w-ordi : Fi → Q such that the composition

λ d λ ji w-ordi Fi → Fi → Q λ λ is the function w-ordi : Fi → Q. d 2. Suppose that the centers Yi of πi+1 are such that Yi ⊂ Max w-ordi . d Then there are functions ti : Fi → Q × Z such that the composition

λ d λ ji ti Fi → Fi → Q × Z λ λ is the function ti : Fi → Q × Z.

Proof. Let E0 = {D1,...,Ds}. The functions ti are completely determined d d by w-ordi , with the constraint that Yi ⊂ Max w-ordi for all i, so we need only d show that w-ordi can be deﬁned for GBOs with d-dimensional structure. λ λ The functions w-ord0 are equal to ord0 , which extend to ord0 on F0 by Proposition 6.27. We thus have deﬁned w-ord0 = ord0. Let Di be the exceptional divisor of π , where i = i + s, and let D˜ λ = W˜ λ · D . i i i i 6.6. Functions on a general basic object 87

For each λ and i we have expressions

cλ cλ (6.28) aλ = I 1 ···I i aλ, i D˜ λ D˜ λ i 1 i λ λ ˜ λ where ai is the weak transform of a0 on Wi , and the cj are locally constant functions on D˜ λ We further have that j

λ 1 λ ai = ai−1OW˜ λ . Ibλ i D˜ λ i

0 Suppose that ηi−1 is a generic point of an irreducible component Yi of λ ˜ λ Yi such that ηi−1 is contained in Wi (and thus necessarily in Wi ), and that qi ∈ Di is a point which we do not require to be closed, but is contained in D˜ λ = W˜ λ · D , and π (q ) is contained in the Zariski closure of η . i i i i i i−1 From (6.28), we deduce that (6.29)  !  Pi−1 λ λ c (ηi−1)νη (I λ ) +νη (a ) λ λ j=1 j i−1 D˜ i−1 i−1 1 c (q ) c (qi) λ  1 i i−1 j λ (a )q = I λ ···I λ I λ a . i i  bλ D˜ D˜ D˜ i  I 1 i−1 i D˜ λ i qi Thus   λ i−1 λ ci (qi) λ X cj (ηi−1) (6.30) λ = w-ordi−1(ηi−1) + λ νηi−1 (ID˜ λ ) − 1. b b j j=1 We further have i 1 X ordλ(q ) = w-ordλ(q ) + cλ(q ). i i i i bλ j j j=1

0 0 0 0 Suppose that qi ∈ Fi, and deﬁne qj for 0 ≤ j ≤ i − 1 by qj = πj+1(qj+1). 0 To simplify notation, we can assume that qj ∈ Yj for all j. 0 0 0 Let Yj be the irreducible component of Yj which contains qj, and let ηj 0 be the generic point of Yj for 0 ≤ j ≤ i − 1. 0 λ Suppose that λ ∈ Λ is such that qi ∈ Wi . We have associated points qj ˜ λ and ηj in Wj for 0 ≤ j ≤ i. We thus have sequences of points ˜ λ {q0, q1, . . . , qi} and {η0, η1, . . . , ηi−1} with qj, ηj ∈ Wj for 0 ≤ j ≤ i and 0 0 0 0 0 0 0 0 λ {q0, q1, . . . , qi} and {η0, η1, . . . , ηi−1} with qj, ηj ∈ Wj for 0 ≤ j ≤ i. 88 6. Resolution of Singularities in Characteristic Zero

From equation (6.30) and induction we see that the values

λ c (qj) (6.31) j for j = 1, . . . , i bλ λ and w-ordi (qi) are determined by: 1. The rational numbers

d 0 d 0 d 0 (6.32) ord0(q0), ord1(q1),..., ordi (qi),

d 0 d 0 d 0 (6.33) ord0(η0), ord1(η1),..., ordi−1(ηi−1). 2. The functions 0 0 1 if qs ∈ Ys and Ys ⊂ Dj locally at qs, (6.34) YH(qs, j) = 0 0 if qs 6∈ Ys or Ys 6⊂ Dj locally at qs. λ Our theorem now follows from Proposition 6.27, since w-ordi (qi) is completely determined by the data (6.32), (6.33) and (6.34).

Proposition 6.29. Let B0 = (F0,W0,E0) be a GBO which admits a d- dimensional structure. Fix notation as in Deﬁnition 6.26. Let

λ λ λ Fi = Sing(ai , b ).

Consider a permissible sequence of transformations

πr π1 Br = (Fr,Wr,Er) → · · · → (F0,W0,E0) such that max w-ordr = 0. Then there are functions

Γr = Γr(Br): Fr → IM = Z × Q × ZN such that the composition

λ λ jr Γr Fr → Fr → IM is the function ˜ λ λ λ λ λ Γ = Γ(Wr , (ar , b ),Er ): Fr → IM of Deﬁnition 6.16.

Proof. Suppose that E0 = {D1,...,Ds}. The function Γ on the basic ˜ λ λ λ λ ci objects (Wr , (ar , b ),Er ) depends only on the values of bλ for 1 ≤ i ≤ r (with the notation of the proof of Theorem 6.28). Thus the conclusions follow from (6.31) of the proof of Theorem 6.28. 6.7. Resolution theorems for a general basic object 89

6.7. Resolution theorems for a general basic object

d d Theorem 6.30. Let (F0 ,W0,E0 ) be a GBO with associated closed sets d {Fi } which admits a d-dimensional structure. Fix notation as in Deﬁnition λ λ λ d ˜ λ 6.26. Let Fi = Sing(ai , b ) = Fi ∩ Wi . Consider a permissible sequence of transformations

d d πr π1 d d (6.35) (Fr ,Wr,Er ) → · · · → (F0 ,W0,E0 ) such that the center Yi of πi+1 satisﬁes

Yi ⊂ Max ti ⊂ Max w-ordi ⊂ Fi for i = 0, 1 . . . , r − 1 and max w-ordr > 0.

This condition implies w-ordi(q) ≤ w-ordi−1(πi(q)), ti(q) ≤ ti−1(πi(q)) for q ∈ Fi and 1 ≤ i ≤ r. Thus

max w-ord0 ≥ max w-ord1 ≥ · · · ≥ max w-ordr,

max t0 ≥ max t1 ≥ · · · ≥ max tr.

Let R(1)(Max tr) be the set of points where Max tr has dimension d − 1. Then R(1) is open and closed in Max tr and non-singular in Wr. d 1. If R(1)(Max tr) 6= ∅, then R(1) is a permissible center for (Fr ,Wr, d Er ) with associated transformation d d πr+1 d d (Fr+1,Wr+1,Er+1) → (Fr ,Wr,Er ),

and either R(1)(Max tr+1) = ∅ or max tr > max tr+1. d−1 d−1 2. If R(1)(Max tr) = ∅, there is a GBO (Fr ,Wr,Er ) with (d−1)- d−1 dimensional structure and associated closed sets {Fi } such that if d−1 d−1 d−1 d−1 (6.36) (FN ,WN ,EN ) → · · · → (Fr ,Wr,Er ) d−1 is a resolution (FN = ∅), then (6.36) induces a permissible sequence of transformations d d d d (FN ,WN ,EN ) → · · · → (Fr ,Wr,Er )

such that the center Yi of πi+1 satisﬁes d Yi ⊂ Max ti ⊂ Max w-ordi ⊂ Fi d−1 for r ≤ i ≤ N − 1, max tr = ··· = max tN−1, Max ti = Fi for i = r, . . . , N − 1, and one of the following holds: d a. FN = ∅. d b. FN 6= ∅ and max w-ordN = 0. d c. FN 6= ∅, max w-ordN > 0 and max tN−1 > max tN . 90 6. Resolution of Singularities in Characteristic Zero

We will now prove Theorem 6.30. Let E0 = {D1,...,Ds}. Let Di be the λ λ exceptional divisor of πi, where i = s + i. Set br = b (max w-ordr) > 0. Let r0 be the smallest integer such that max w-ordr0 = max w-ordr in (6.35). λ There exist an open cover {W0 } of W0 and a d-dimensional structure λ λ ˜ λ λ λ ˜λ {B0 }λ∈Λ of (F0,W0,E0), where B0 = (W0 , (a0 , b ), E0 ) for λ ∈ Λ. Thus for λ ∈ Λ, we have sequences of transformations ˜ λ λ λ ˜λ πr π1 ˜ λ λ λ ˜λ (Wr , (ar , b ), Er ) → · · · → (W0 , (a0 , b ), E0 ), λ ˜ λ ˜ λ where the center Yi of Wi+1 → Wi is contained in Max ti. There is an expression cλ λ aλ = I 1 ···Icr aλ. r Dλ Dλ r 1 r We now deﬁne a basic object 0 λ ˜ λ 0 λ 0 λ ˜λ (Br) = (Wr , ((ar) , (b ) ), Er ) ˜ λ on Wr by ( aλ if bλ ≥ bλ, 0 λ r r (6.37) (a ) = λ λ cλ cλ λ r (aλ)b −br + (I 1 ···I r )br if bλ < bλ, r Dλ Dλ r 1 r

λ λ λ 0 λ br if br ≥ b , (b ) = λ λ λ λ λ br (b − br ) if br < b . Lemma 6.31. 0 λ 0 λ λ 1. Sing((ar) , (b ) ) = (Max w-ordr) ∩ Wr . ˜ λ 0 λ 0 λ ˜λ 2. (Wr , ((ar) , (b ) ), Er ) is a simple basic object. 3. A transformation ˜ λ 0 λ 0 λ ˜λ ˜ λ 0 λ 0 λ ˜λ (Wr+1, ((ar+1) , (b ) ), Er+1) → (Wr , ((ar) , (b ) ), Er ) λ with center Yr induces a transformation ˜ λ λ λ ˜λ ˜ λ λ λ ˜λ (Wr+1, (ar+1, b ), Er+1) → (Wr , (ar , b ), Er ) ˜ λ ˜ λ such that the center Yr ⊂ Max w-ordr ∩Wr . 0 λ 0 λ 4. Sing((ar+1) , (b ) ) = ∅ if and only if one of the following holds: a. max w-ordr > max w-ordr+1, or λ b. max w-ordr = max w-ordr+1 and (Max w-ordr) ∩ Wr = ∅. 5. If max w-ordr = max w-ordr+1 then 0 λ 0 λ ˜ λ Sing((ar+1) , (b ) ) = Max w-ordr+1 ∩Wr+1, 0 λ λ and (ar+1) is deﬁned in terms of ar+1 by (6.37). 6.7. Resolution theorems for a general basic object 91

˜ λ λ λ Proof. 1. For q ∈ Wr we have q ∈ Max w-ordr if and only if νq(ar ) ≥ br λ λ 0 λ 0 λ and νq(ar ) ≥ b . These conditions hold if and only if νq((ar) ) ≥ (b ) . λ λ ˜ λ 2 follows since νq(ar ) ≤ br for q ∈ Wr . 3 is immediate from 1. λ λ If br ≥ b , we have λ 0 λ (6.38) aλ = I−br a = I−(b ) (a0 )λ = (a0 )λ. r+1 Dλ r Dλ r r+1 r+1 r+1 λ λ Suppose that br < b . Then λ λ br λ λ cλ c (aλ )b −br + I 1 ···I r+1 r+1 Dλ Dλ 1 r+1 (6.39) λ λ br −bλ λ λ cλ c = (I r a )b −br + I 1 ···I r+1 , Dλ r Dλ Dλ r+1 1 r+1 where r λ X λ λ cr+1(q) = cj(η)νη(IDλ ) + br − b j j=1 ˜ λ if η is the generic point of the component of Yr containing πr+1(q). cλ cλ If we set K = I 1 ···I r+1 (the ideal sheaf on W˜ λ), we see that (6.39) Dλ Dλ r 1 r+1 is equal to

λ λ λ (6.40) (a + K)I(b −br )br = (a0 )λ. r Dλ r+1 r+1 Now 4 and 5 follow from 1 and equations (6.38), (6.39) and (6.40). At q ∈ Max t ∩ W˜ λ there exist N = η(q) hypersurfaces Dλ ,...,Dλ ∈ r r i1 iN ˜− λ (Er ) such that λ λ q ∈ Di1 ∩ · · · ∩ Din ∩ Max w-ordr 6= ∅. λ,φ Thus there exists an aﬃne neighborhood Ur of q such that (6.41) Max t ∩ U λ,φ = Dλ,φ ∩ · · · ∩ Dλ,φ ∩ Max w-ordλ,φ r r i1 in r λ,φ (The superscript λ, φ denotes restriction to Ur .) Now deﬁne a basic object 00 λ,φ λ,φ 00 λ,φ 00 λ,φ ˜+ λ,φ (Br ) = (Ur , ((ar ) , (b ) ), (Er ) ) by 00 λ,φ 0 λ,φ (b00)λ,φ (b00)λ,φ (ar ) = (ar) + (IDλ,φ ) + ··· + (IDλ,φ ) , (6.42) i1 iN (b00)λ,φ = (b0)λ. 92 6. Resolution of Singularities in Characteristic Zero

Lemma 6.32. 00 λ,φ 00 λ,φ λ,φ 1. Sing((ar ) , (b ) ) = Max tr ∩ Ur . λ,φ 00 λ,φ 00 λ,φ ˜ λ,φ 2. (Ur , ((ar ) , (b ) ), (Er+) ) is a simple basic object. 3. A transformation λ,φ 00 λ,φ 00 λ,φ ˜+ λ,φ λ,φ 00 λ,φ 00 λ,φ ˜+ λ,φ (Ur+1, ((ar+1) , (b ) ), (Er+1) ) → (Ur , ((ar ) , (b ) ), (Er ) ) λ,φ with center Yr induces a transformation λ,φ λ,φ λ ˜λ,φ λ,φ λ,φ λ ˜λ,φ (Ur+1, (ar+1, b ), Er+1) → (Ur , (ar , b ), Er ) λ,φ such that Yr ⊂ Max tr. 00 λ,φ 00 λ,φ 4. Sing((ar+1) , (b ) ) = ∅ if and only if one of the following conditions holds: a. max tr > max tr+1, or λ,φ b. max tr = max tr+1 and Max tr+1 ∩ Ur+1 = ∅. λ 5. If max tr = max tr+1 and Max tr+1 ∩ Wr+1 6= ∅, then 00 λ,φ 00 λ,φ λ,φ Sing((ar+1) , (b ) ) = Max w-ordr+1 ∩Ur+1 00 λ,φ λ 0 λ and (ar+1) is deﬁned in terms of ar+1 and (ar+1) by (6.35) and (6.37).

λ,φ Proof. 1. For q ∈ U1 we have q ∈ Max tr if and only if q ∈ Max w-ordr 0 λ 0 λ and η(q) = N, which hold if and only if νq((ar) ) ≥ (b ) (by 1 of Lemma λ,φ 6.31) and νq(I ) ≥ 1 for 1 ≤ j ≤ N, which holds if and only if q ∈ Dij 00 λ,φ 00 λ,φ Sing((ar ) , (b ) ). 00 λ,φ 00 λ,φ λ,φ 2 follows since νq((ar ) ) ≤ (b ) for q ∈ Ur . 3 is immediate from 1. 4 and 5 follow from 4 and 5 of Lemma 6.31, and the observation that if λ,φ max tr+1 = max tr and q ∈ Max tr+1 ∩ Ur+1, then q must be on the strict transforms of the hypersurfaces Dλ,φ,...,Dλ,φ. i1 iN We now observe that the conclusions of Lemmas 6.31 and 6.32, which are formulated for transformations, can be naturally formulated for restrictions 00 λ,φ also. Then it follows that the (Br ) deﬁne a GBO with d-dimensional d + structure on (Wr, (Er ) ): 00 d + (6.43) (Fr ,Wr, (Er ) ), 00 with associated closed sets Fi = Max ti. If Y is a permissible center for (6.43), then Y makes simple normal + crossings with Er . The local description (6.41) and (6.42) then implies that 6.7. Resolution theorems for a general basic object 93

Y makes simple normal crossings with Er. Thus Y ⊂ Max tr is a permissible d d center for (Fr ,Wr,Er ). If the induced transformations are 00 d + 00 d + (Fr+1,Wr+1, (Er+1) ) → (Fr ,Wr, (Er ) ) and d d d d (Fr+1,Wr+1,Er+1) → (Fr ,Wr,Er ), 00 then either max tr > max tr+1, in which case Fr+1 = ∅, or max tr = max tr+1 00 and Fr+1 = Max tr+1. We will now verify that the assumptions of Theorem 6.24 hold for the λ,φ 00 λ,φ 00 λ,φ ˜+ λ,φ simple basic object (Ur , ((ar ) , (b ) ), (Er ) ). 0 λ 0 λ Define (by (6.37)) (Br0 ) in the same way that we defined (Br) . Now 0 λ 5 of Lemma 6.31 implies (Br) is obtained by a sequence of transformations 0 λ λ + of the simple basic object (Br0 ) .(Er ) is the exceptional divisor of the product of these transformations. 0 λ λ,Ψ Since (Br0 ) is a simple basic object, we can find an open cover {Vr0 } ˜ λ λ,Ψ λ,Ψ of Wr0 with non-singular hypersurfaces (Vh)r0 in Vr0 such that (b0)λ−1 0 λ,Ψ λ,Ψ I ⊂ 4 ((ar0 ) ). (Vh)r0 λ,Ψ λ,ψ λ,ψ Let (Vh)r be the strict transform of (Vh)r0 in Vr . By Lemma 6.21, λ,ψ (Vh)r is non-singular, (b0)λ−1 0 λ,ψ I λ,ψ ⊂ 4 ((a ) ), (Vh)r r λ,ψ + λ,ψ λ,φ,ψ λ,ψ and we have that (Vh)r makes SNCs with (Er ) . Let Ur = Vr ∩ λ,φ λ,φ,ψ λ,φ λ,φ,ψ Ur . For fixed λ and φ, {Ur } is an open cover of Ur . For q ∈ Ur ,

(b0)λ−1 0 λ,ψ (b00)λ,φ−1 00 λ,φ (I λ,ψ )q ⊂ 4 ((ar) ) ⊂ 4 (ar ) (Vh)r q q by (6.42). Thus the assumptions of Theorem 6.24 hold for 00 λ,φ λ,φ 00 λ,φ 00 λ,φ ˜+ λ,ψ (Br ) = (Ur , ((ar ) , (b ) ), (Er ) ), λ,φ,ψ and {Ur }. Assertion 1 of Theorem 6.30 now follows from Theorem 6.24 00 λ,φ applied to our GBO of (6.43), with d-dimensional structure (Br ) . If R(1)(Max tr) is empty, then by Theorem 6.24, applied to the GBO of (6.43), and its d-dimensional structure, (6.43) has a (d − 1)-dimensional structure, and we set d−1 d−1 00 d + (Fr ,Wr,Er ) = (Fr ,Wr, (Er ) ). Given a resolution (6.36), by Lemma 6.32, we have an induced sequence d d d d (FN ,WN ,EN ) → · · · → (Fr ,Wr,Er ) such that the conclusions of 2 of Theorem 6.30 hold. 94 6. Resolution of Singularities in Characteristic Zero

Lemma 6.33. Let I1 and I2 be totally ordered sets. Suppose that

πN π1 (6.44) (WN ,EN ) → · · · → (W0,E0) is a sequence of transformations of pairs together with closed sets Fi ⊂ Wi such that the center Yi of πi+1 is contained in Fi for all i, πi+1(Fi+1) ⊂ Fi for all i, and FN = ∅. Assume that for 0 ≤ i ≤ N the following conditions hold:

1. There is an upper semi-continuous function gi : Fi → I1. 0 2. There is an upper semi-continuous function gi : Max gi → I2. 0 3. Yi = Max gi in (6.44). 4. For any q ∈ Fi+1, with 0 ≤ i ≤ N − 2,

gi(πi+1(q)) ≥ gi+1(q) if πi+1(q) ∈ Yi, gi(πi+1(q)) = gi+1(q) if πi+1(q) 6∈ Yi. This condition implies that

max g0 ≥ max g1 ≥ · · · ≥ max gN−1,

and if max gi = max gi+1, then πi+1(Max gi+1) ⊂ Max gi.

5. If max gi = max gi+1 and q ∈ Max gi+1, with 0 ≤ i ≤ N − 2, then 0 0 gi(πi+1(q)) > gi+1(q) if πi+1(q) ∈ Yi, 0 0 gi(πi+1(q)) = gi+1(q) if πi+1(q) 6∈ Yi. Then there are upper semi-continuous functions

fi : Fi → I1 × I2 deﬁned by 0 (max gi, max gi) if q ∈ Yi, (6.45) fi(q) = fi+1(q) if q 6∈ Yi, where in the second case we can view q as a point on Wi+1, as Wi+1 → Wi is an isomorphism in a neighborhood of q. For q ∈ Fi+1,

fi(πi+1(q)) > fi(q) if πi+1(q) ∈ Yi, (6.46) fi(πi+1(q)) = fi+1(q) if πi+1(q) 6∈ Yi, 0 0 max fi = (max gi, max gi) and Max fi = Max gi.

Proof. We first show, by induction on i, that the fi of (6.45) are well defined functions. Since FN = ∅, YN−1 = FN−1. By assumption 3, 0 Max gN−1 = Max gN−1 = FN−1, so we can define 0 0 fN−1(q) = (gN−1(q), gN−1(q)) = (max gN−1, max gN−1) for q ∈ FN−1. 6.7. Resolution theorems for a general basic object 95

Suppose that q ∈ Fr. If q ∈ Yr, we deﬁne 0 0 fr(q) = (gr(q), gr(q)) = (max gr, max gr).

Suppose that q 6∈ Yr. As πi+1(Fi+1) ⊂ Fi for all i, we have an isomorphism ∼ Fr − Yr = Fr+1 − Dr+1, where Dr+1 is the reduced exceptional divisor of Wr+1 → Wr. Since FN = ∅, we can deﬁne fr(q) = fr+1(q) for q ∈ Fr − Yr. The properties (6.46) are immediate from the assumptions and (6.45).

It remains to prove that fr is upper semi-continuous. We prove this by descending induction on r. Fix α = (α1, α2) ∈ I1 × I2. We must prove that {q ∈ Fr | fr(q) ≥ α} is closed. If max fr < α, then the set is empty. If max fr ≥ α, then 0 0 0 {q ∈ Fr | fr(q) ≥ α} = Max gr ∪ πr+1({q ∈ Fr+1 | fr+1(q ) ≥ α}), which is closed by upper semi-continuity of fr+1 and properness of πr+1.

Theorem 6.34. Fix an integer d ≥ 0. There are a totally ordered set Id d and functions fi with the following properties: d d 1. For each GBO (F0 ,W0,E0 ) with a d-dimensional structure and d associated closed subsets Fi there is a function d d f0 : F0 → Id d d with the property that Max f0 is a permissible center for (W0,E0 ). 2. If a sequence of transformations with permissible centers Yi d d d d (6.47) (Fr ,Wr,Er ) → · · · → (F0 ,W0,E0 ) d d and functions fi : Fi → Id, i = 0, . . . , r − 1, have been deﬁned with d d d the property that Yi = Max fi , then there is a function fr : Fr → Id d d such that Max fr is permissible for (Wr,Er ). d d 3. For each GBO (F0 ,W0,E0 ) with a d-dimensional structure, there is an index N so that the sequence of transformations d d d d (6.48) (FN ,WN ,EN ) → · · · → (F0 ,W0,E0 ).

constructed by 1 and 2 is a resolution (FN = ∅). d 4. The functions fi in 2 have the following properties: d a. If q ∈ Fi with 0 ≤ i ≤ N − 1, and if q 6∈ Yi, then fi (q) = d fi+1(q). d b. Yi = Max fi for 0 ≤ i ≤ N − 1, and d d d max f0 > max f1 > ··· > max fN−1 d c. For 0 ≤ i ≤ N − 1 the closed set Max fi is smooth, equidimensional, and its dimension is determined by the value of d max fi . 96 6. Resolution of Singularities in Characteristic Zero

Proof. We prove Theorem 6.34 by induction on d.

First assume that d = 1. Set I1 to be the disjoint union I1 = Q×Zt{∞}, where Q × Z is ordered lexicographically, and ∞ is the maximum of I1. 1 1 Suppose that (F0 ,W0,E0 ) is a GBO with one-dimensional structure, and 1 1 1 1 with associated closed sets Fi . Define f0 = t0 (t0 is defined by Theorem 1 6.28). The closed set F0 is (locally) a codimension ≥ 1 subset of a one- 1 dimensional subvariety of W0, so dim F0 = 0. Thus R(1)(Max t0) 6= ∅, and we are in the situation of 1 of Theorem 6.30. If we perform the permissible 1 transformation with center Max t0, 1 1 1 1 (F1 ,W1,E1 ) → (F0 ,W0,E0 ), 1 1 we have max t0 > max t1. We can thus define a sequence of transformations 1 1 1 1 (Fr ,Wr,Er ) → · · · → (F0 ,W0,E0 ), 1 1 1 1 where each transformation has center Max ti , fi = ti : Fi → I1 and 1 1 max t0 > ··· > max tr. 1 1 Since there is a natural number b such that max ti ∈ b! Z × Z for all i, there is an r such that the above sequence is a resolution. Now assume that d > 1 and that the conclusions of Theorem 6.34 hold for GBOs of dimension d − 1. Thus there are a totally ordered set Id−1 and d−1 functions fi satisfying the conclusions of the theorem. 0 Let Id be the disjoint union 0 Id = Q × Z t IM t {∞}, 0 where IM is the ordered set of Definition 6.16. We order Id as follows: Q×Z has the lexicographic order. If α ∈ Q × Z and β ∈ IM , then β < α, and ∞ 0 0 is the maximal element of Id. Define Id = Id × Id−1 with the lexicographic ordering. d d Suppose that (F0 ,W0,E0 ) is a GBO of dimension d, with associated d d 0 closed sets Fi . Define g0 : F0 → Id by d g0(q) = t0(q), 0 and g0 : Max g0 → Id−1 by d 0 ∞ if q ∈ R(1)(Max t0), g0(q) = d−1 d f0 (q) if q 6∈ R(1)(Max t0), d−1 where f0 is the function defined by the GBO of dimension d − 1 d−1 d−1 (F0 ,W0,E0 ) of 2 of Theorem 6.30. 6.7. Resolution theorems for a general basic object 97

Assume that we have now inductively deﬁned a sequence of transformations

d d πr π1 d d (6.49) (Fr ,Wr,E1 ) → · · · → (F0 ,W0,E0 ) and we have defined functions d 0 0 gi : Fi → Id and gi : Max gi → Id−1 0 satisfying the assumptions 1-5 of Lemma 6.33 (with I1 = Id and I2 = Id−1) 0 for i = 0, . . . , r − 1. In particular, the center of πi+1 is Yi = Max gi. d d 0 If Fr 6= ∅, define gr : Fr → Id by d Γ(Br)(q) if w-ordr (q) = 0, gr(q) = d d tr (q) if w-ordr (q) > 0, where Γ(Br) is the function defined in Proposition 6.29 for the GBO Br = d d 0 (Fr ,Wr,Er ). Define gr : Max gr → Id−1 by  d ∞ if w-ordr (q) = 0, 0  d gr(q) = ∞ if q ∈ R(1)(Max tr ) and w-ordr(q) > 0,  d−1 d fr (q) if q 6∈ R(1)(Max tr ) and w-ordr(q) > 0, d−1 where fr is the function defined by the GBO of dimension d − 1 d−1 d−1 (Fr ,Wr,Er ) in 2 of Theorem 6.30. Observe that if max w-ordr > 0, then the center Yi of 0 d πi+1 in (6.49) is defined by Yi = Max gi ⊂ Max ti for 0 ≤ i ≤ r − 1. Thus (if max w-ordr > 0) we have that (6.49) is a sequence of the form of (6.35) of Theorem 6.30. Now Theorem 6.30, our induction assumption, and Lemma 6.17 show that the sequence (6.49) extends uniquely to a resolution

(FN ,WN ,EN ) → · · · → (F0,W0,E0) 0 with functions gi and gi satisfying assumptions 1-5 of Lemma 6.33. By 0 Lemma 6.33, the functions (gi, gi) extend to functions d 0 Fi → Id = Id × Id−1 satisfying the conclusions of Theorem 6.34. d Consider the values of the functions fi which we constructed at a point d q ∈ Fi. fi (q) has d coordinates, and has one of the following three types: d d d−1 d−r 1. fi (q) = (t (q), t (q), . . . , t (q), ∞, ··· , ∞). d d d−1 d−r 2. fi (q) = (t (q), t (q), . . . , t (q), Γ(q), ∞, ··· , ∞). d d d−1 1 3. fi (q) = (t (q), t (q), . . . , t (q)). 98 6. Resolution of Singularities in Characteristic Zero

Each coordinate is a function defined for a GBO of the corresponding dimension. We have ti(q) = (w-ordi(q), ηi(q)) with w-ordi(q) > 0. In case 1, td−r is such that q ∈ R(1)(Max td−r); that is, Max td−r has d codimension 1 in a d − r dimensional GBO, and dim(Max fi ) = d − r − 1. In case 2, w-ordd−(r+1)(q) = 0 and Γ is the function defined in Proposi- d tion 6.29 for a monomial GBO and dim(Max fi ) = d − r − Γ1 − 1, where Γ1 is the first coordinate of max Γ. d In case 3, dim(Max fi ) = 0. d Thus Max fi is non-singular and equidimensional, and 4 c follows. Theorem 6.35. Suppose that d is a positive integer. Then there are a 0 d totally ordered set Id and functions pi with the following properties:

1. For each pair (W0,E0) with d = dim W0 and each ideal sheaf J0 ⊂

OW0 there is a function d 0 p0 : V (J0) → Id d with the property that Max p0 ⊂ V (J0) is permissible for (W0,E0). 2. If a sequence of transformations of pairs with centers Yi ⊂ V (J i)

(Wr,Er) → · · · → (W0,E0) d 0 and functions pi : V (J i) → Id, for 0 ≤ i ≤ r − 1, has been deﬁned d with the property that Yi = Max pi and V (J k) 6= ∅, then there is d 0 d a function pr : V (J r) → Id such that Max pr is permissible for (Wr,Er).

3. For each pair (W0,E0) and J0 ⊂ OW0 there is an index N such that the sequence of transformations

(WN ,EN ) → · · · → (W0,E0)

constructed inductively by 1 and 2 is such that V (J N ) = ∅. The corresponding sequence will be called a “strong principalization” of J0. 4. Property 4 of Theorem 6.34 holds.

Proof. Let J = J , and set b = max ν . By Theorem 6.34, there exists 0 0 0 J0 a resolution

(WN1 , (J N1 , b0),EN1 ) → · · · → (W0, (J 0, b0),E0) of the simple basic object (W0, (J 0, b0),E0). Thus we have that J N1 (which is the weak transform of J0 on OWN ) satisﬁes b1 = max ν < b0. We now 1 JN1 can construct by Theorem 6.34 a resolution

(WN2 , (J N2 , b1),EN2 ) → · · · → (WN1 , (J N1 , b1),EN1 ) 6.8. Resolution of singularities in characteristic zero 99

of the simple basic object (WN1 , (J N1 , b1),EN1 ). After constructing a ﬁnite number of resolutions of simple basic objects in this way, we have a sequence of transformations of pairs

(6.50) (WN ,EN ) → · · · → (W0,E0) such that the strict transform J N of J0 on WN is J N = OWN . We have upper semi-continuous functions f d : Max ν → I i Ji d satisfying the conclusions of Theorem 6.34 on the resolution sequences

(WNi+1 , (J Ni+1 , bi),ENi+1 ) → · · · → (WNi , (J Ni , bi),ENi ).

Now apply Lemma 6.33 to the sequence (6.50), with Fi = V (J i), gi = ν , g0 = f d. We conclude that there exist functions pd : V (J ) → I0 = ×I Ji i i i i d Z d with the desired properties.

6.8. Resolution of singularities in characteristic zero

Recall our conventions on varieties in the notation part of Section 1.1. Theorem 6.36 (Principalization of Ideals). Suppose that I is an ideal sheaf on a non-singular variety W over a ﬁeld of characteristic zero. Then there exists a sequence of monodial transforms

π : W1 → W which is an isomorphism away from the closed locus of points where I is not locally principal, such that IOW1 is locally principal.

Proof. We can factor I = I1I2, where I1 is an invertible sheaf and V (I2) is the set of points where I is not locally principal. Then we apply Theorem 6.35 to J0 = I2. Theorem 6.37 (Embedded Resolution of Singularities). Suppose that X is an algebraic variety over a ﬁeld of characteristic zero which is embedded in a non-singular variety W . Then there exists a birational projective morphism

π : W1 → W such that π is a sequence of monodial transforms, π is an embedded resolution of X, and π is an isomorphism away from the singular locus of W .

Proof. Set X0 = X, W0 = W , J0 = IX0 ⊂ OW0 . Let

(WN ,EN ) → (W0,E0) 100 6. Resolution of Singularities in Characteristic Zero

be the strong principalization of J0 constructed in Theorem 6.35, so that the weak transform of J0 on WN is J N = OWN . With the notation of the proofs of Theorem 6.34 and Theorem 6.35, if X0 has codimension r in the d-dimensional variety W0, then for q ∈ Reg(X0), d 0 p0(q) = (1, (1, 0),..., (1, 0), 0,..., 0) ∈ Id, d where there are r copies of (1, 0) followed by d − r zeros. The function p0 is thus constant on the non-empty open set Reg(X0) of non-singular points 0 of X0. Let this constant be c ∈ Id. By property 4 of Theorem 6.34 there d exists a unique index r ≤ N − 1 such that max pr = c. Since Wr → W0 is an isomorphism over the dense open set Reg(X0), the strict transform Xr of d X0 must be the union of the irreducible components of the closed set Max pr d of Wr. Since Max pr is a permissible center, Xr is non-singular and makes simple normal crossings with the exceptional divisor Er. Theorem 6.38 (Resolution of Singularities). Suppose that X is an algebraic variety over a ﬁeld of characteristic zero. Then there is a resolution of singularities π : X1 → X such that π is a projective morphism which is an isomorphism away from the singular locus of X.

Proof. This is immediate from Theorem 6.37, after choosing an embedding of X into a projective space W . Theorem 6.39 (Resolution of Indeterminacy). Suppose that K is a ﬁeld of characteristic zero and φ : W → V is a rational map of projective K- varieties. Then there exist a projective birational morphism π : W1 → W such that W1 is non-singular and a morphism λ : W1 → V such that λ = φ ◦ π. If W is non-singular, then π is a product of monoidal transforms.

Proof. By Theorem 6.38, there exists a resolution of singularities ψ : W1 → W . After replacing W with W1 and φ with ψ ◦ φ, we may assume that W is non-singular. Let Γφ be the graph of φ, with projections π1 :Γφ → W and π2 :Γφ → V . As π1 is birational and projective, there exists an ideal ∼ sheaf I ⊂ OW such that Γφ = B(I) is the blow-up of I (Theorem 4.5). By Theorem 6.36, there exists a principalization π : W1 → W of I. The universal property of blowing up (Theorem 4.2) now shows that there is a morphism λ : W1 → V such that λ = φ ◦ π. Theorem 6.40. Suppose that π : Y → X is a birational morphism of projective non-singular varieties over a ﬁeld K of characteristic 0. Then i ∼ i H (Y, OY ) = H (X, OX ) ∀ i. 6.8. Resolution of singularities in characteristic zero 101

Proof. By Theorem 6.39, there exists a projective morphism f : Z → Y such that g = π ◦ f is a product of blow-ups of non-singular subvarieties,

gn gn−1 g2 g1 g : Z = Zn → Zn−1 → ... → Z1 → Z0 = X. We have ([65] or Lemma 2.1 [31]) i 0, i > 0, R gj∗OZj = OZj−1 , i = 0. Thus, by the Leray spectral sequence, i 0, if i > 0, (6.51) R g∗OZ = OX , if i = 0, and ∗ i ∼ i g : H (X, OX ) = H (Z, OZ ) for all i. Since g∗ = f ∗ ◦π∗, we conclude that π∗ is one-to-one. To show that π∗ is an isomorphism we now only need to show that f ∗ is also one-to-one. Theorem 6.40 also gives a projective morphism γ : W → Z such that β = f ◦ γ is a product of blow-ups of non-singular subvarieties, so we have ∗ i ∼ i β : H (Y, OY ) = H (W, OW ) ∗ for all i. This implies that f is one-to-one, and the theorem is proved. For the following theorem, which is stronger than Theorem 6.38, we give a complete proof only the case of a hypersurface. In the embedded resolution constructed in Theorem 6.37, which induces the resolution of 6.38, if X is not a hypersurface, there may be monodial transforms Wi+1 → Wi whose centers are not contained in the strict transform Xi of X, so that the induced morphism Xi+1 → Xi of strict transforms of X will in general not be a monodial transform. Theorem 6.41 (A Stronger Theorem of Resolution of Singularities). Sup- pose that X is an algebraic variety over a ﬁeld of characteristic zero. Then there is a resolution of singularities

π : X1 → X such that π is a sequence of monodial transforms centered in the closed sets of points of maximum multiplicity.

Proof. In the case of a hypersurface X = X0 embedded in a non-singular variety W = W0, the proof given for Theorem 6.37 actually produces a resolution of the kind asserted by this theorem, since the resolution sequence is constructed by patching together resolutions of the simple basic objects (Wi, (J i, b),Ei), where J i is the weak transform of the ideal sheaf J0 of X0 in W0. Since J0 is a locally principal ideal, and each transformation in 102 6. Resolution of Singularities in Characteristic Zero

these sequences is centered at a non-singular subvariety of Max J i, the weak transform J i is actually the strict transform of J0. For the case of a general variety X, we choose an embedding of X in a projective space W . We then modify the above proof by considering the Hilbert-Samuel function of X. It can be shown that there is a simple basic object (W0, (K0, b),E0) such that Max νK0 is the maximal locus of d the Hilbert-Samuel function. In the construction of pi of Theorem 6.35 we use ν in place of ν . Then the resolution of this basic object induces a K0 Ji sequence of monodial transformations of X such that the maximum of the Hilbert-Samuel function has dropped. More details about this process are given, for instance, in [55], [14] and [79]. Exercise 6.42. 1. Follow the algorithm of this chapter to construct an embedded resolution of singularities of: 2 3 2 a. The singular curve y − x = 0 in A . 2 2 3 3 b. The singular surface z − x y = 0 in A . 2 3 3 c. The singular curve z = 0, y − x = 0 in A . d 2. Give examples where the fi achieve the 3 cases of the proof of Theorem 6.34. 3. Suppose that K is a field of characteristic zero and X is an integral finite type K-scheme. Prove that there exists a proper birational morphism π : Y → X such that Y is non-singular. 4. Suppose that K is a field of characteristic zero and X is a reduced finite type K-scheme. Prove that there exists a proper birational morphism π : Y → X such that Y is non-singular. 5. Let notation be as in the statement of Theorem 6.35. a. Suppose that q ∈ V (J0) ⊂ W0 is a closed point. Let U =

spec(OW0,q), Di = Di ∩U for 1 ≤ i ≤ r and E = {D1,..., Dr}. d d Consider the function p0 of Theorem 6.35. Show that p0(q) depends only on the local basic object (U, E) and the ideal d (J0)q ⊂ OW0,q. In particular, p0(q) is independent of the K- algebra isomorphism of OX,q which takes J0OX,q to J0OX,q and Di to Di for 1 ≤ i ≤ r. b. Suppose that Θ : W0 → W0 is a K-automorphism such that ∗ d d Θ (J0) = J0. Show that p0(Θ(q)) = p0(q) for all q ∈ V (J0). c. Suppose that Θ : W0 → W0 is a K-automorphism, and Y ⊂ W0 is a non-singular subvariety such that Θ(Y ) = Y . Let π1 : W1 → W be the monoidal transform centered at Y . Show that Θ extends to a K-automorphism Θ : W1 → W1 such that 6.8. Resolution of singularities in characteristic zero 103

Θ(π1(q)) = π1(Θ(q)) for all q ∈ W1 and Θ(D) = D if D is the exceptional divisor of π1.

6. Suppose that (W0,E0) is a pair and J0 ⊂ OW0 is an ideal sheaf, with notation as in Theorem 6.35. Consider the sequence in 2 of Theorem 6.35. Suppose that G is a group of K-automorphisms of ∗ W0 such that Θ(Di) = Di for 1 ≤ i ≤ r and Θ (J0) = J0. Show that G extends to a group of K-automorphisms of each Wi in the sequence 2 of Theorem 6.35 such that Θ(Yi) = Yi for all Θ ∈ G and each πi is G-equivariant. That is,

πi(Θ(q)) = Θ(πi(q)) d for all q ∈ Wi. Furthermore, the functions pi of Theorem 6.35 d d satisfy pi (Θ(q)) = pi (q) for all q ∈ Wi. 7. (Equivariant resolution of singularities) Suppose that X is a variety over a ﬁeld K of characteristic zero. Suppose that G is a group of K-automorphisms of X. Show that there exists an equivariant resolution of singularities π : X1 → X of X. That is, G extends to a group of K-automorphisms of X so that π(Θ(q)) = Θ(π(q)) for all q ∈ X1. More generally, deduce equivariant versions of all the other theorems of Section 6.8, and of parts 3 and 4 of Exercise 6.42).

Chapter 7

Resolution of Surfaces in Positive Characteristic

7.1. Resolution and some invariants

In this chapter we prove resolution of singularities for surfaces over an algebraically closed ﬁeld of characteristic p ≥ 0. The characteristic 0 proof in Section 5.1 depended on the existence of hypersurfaces of maximal contact, which is false in positive characteristic (see Exercise 7.35 at the end of this section). The algorithm of this chapter is sketched by Hironaka in his notes [53]. We prove the following theorem: Theorem 7.1. Suppose that S is a surface over an algebraically closed ﬁeld K of characteristic p ≥ 0. Then there exists a resolution of singularities Λ: T → S.

Theorem 7.1 in the case when S is a hypersurface follows from induction on r = max{t | Singt(S) 6= 0} in Theorems 7.6, 7.7 and 7.8. We then obtain Theorem 7.1 in general from Theorem 4.9 and Theorem 4.11. For the remainder of this chapter we will assume that K is an algebraically closed ﬁeld of characteristic p ≥ 0, V is a non-singular 3-dimensional variety over K, and S is a surface in V . It follows from Theorem A.19 that for t ∈ N, the set

Singt(S) = {p ∈ S | νp(S) ≥ t} is Zariski closed.

105 106 7. Resolution of Surfaces in Positive Characteristic

Let r = max{t | Singt(S) 6= ∅}.

Suppose that p ∈ Singr(S) is a closed point, f = 0 is a local equation of S at p, and (x, y, z) are regular parameters at p in V (or in OˆV,p). There is an expansion X i j k f = aijkx y z i+j+k≥r with aijk ∈ K in OˆV,p = K[[x, y, z]]. The leading form of f is deﬁned to be X i j k L(x, y, z) = aijkx y z . i+j+k=r We deﬁne a new invariant, τ(p), to be the dimension of the smallest linear subspace M of the K-subspace spanned by x, y and z in K[x, y, z] such that L ∈ k[M]. This subspace is uniquely determined. This dimension is in fact independent of the choice of regular parameters (x, y, z) at p (or in OˆV,p). If x, y, z are regular parameters in OV,p, we will call the subvariety N = V (M) of spec(OV,p) an “approximate manifold” to S at p. If (x, y, z) are regular parameters in OˆV,p, we call N = V (M) ⊂ spec(OˆV,p) a (formal) approximate manifold to S at p. M is dependent on our choice of regular parameters at p. Observe that 1 ≤ τ(q) ≤ 3.

If there is a non-singular curve C ⊂ Singr(S) such that q ∈ C, then τ(q) ≤ 2. Example 7.2. Let f = x2 + y2 + z2 + x5. If char(K) 6= 2, then τ = 3 and x = y = z = 0 are local equations of the approximate manifold at the origin. However, if char(K) = 2, then τ = 1 and x+y +z = 0 is a local equation of an approximate manifold at the origin. Example 7.3. Let f = y2 + 2xy + x2 + z2 + z5, char(K) > 2. Then L = (y + x)2 + z2, so that τ = 2 and x + y = z = 0 are local equations of an approximate manifold at the origin.

Lemma 7.4. Suppose that q ∈ S is a closed point such that νq(S) = r, τ(q) = n (with 1 ≤ n ≤ 3), and L1 = ··· = Ln = 0 are local equations at q of a (possibly formal) approximate manifold N of S at q. Let π : V1 → V −1 be the blow-up of q, E = π (q) the exceptional divisor of π, S1 the strict transform of S on V1, and N1 the strict transform of N on V1. Suppose that −1 q1 ∈ π (q). Then νq1 (S1) ≤ νq(S), and if νq1 (S1) = r, then τ(q) ≤ τ(q1) and q1 ∈ N1 ∩ E.

Moreover, N1 ∩ E = ∅ if τ(q) = 3, N1 ∩ E is a point if τ(q) = 2, and N1 ∩ E is a line if τ(q) = 1. 7.1. Resolution and some invariants 107

Proof. Let f = 0 be a local equation of S at q, and let (x, y, z) be regular parameters at p such that 1. x = y = z = 0 are local equations of N if τ(q) = 3, 2. y = z = 0 are local equations of N if τ(q) = 2, 3. z = 0 is a local equation of N if τ(q) = 1.

In OˆV,p = K[[x, y, z]], write X i j k f = aijkx y z = L + G, where L is the leading form of degree r, and G has order > r. q1 has regular parameters (x1, y1, z1) such that one of the following cases holds:

1. x = x1, y = x1(y1 + α), z = x1(z1 + β) with α, β ∈ K.

2. x = x1y1, y = y1, z = y1(z1 + β) with β ∈ K.

3. x = x1z1, y = y1z1, z = z1.

Suppose that the ﬁrst case holds. S1 has a local equation f1 = 0 at q1 such that f1 = L(1, y1 + α, z1 + β) + x1Ω.

Thus νq1 (f1) ≤ r, and νq1 (f1) = r implies X j k X j k L(1, y1 + α, z1 + β) = aijk(y1 + α) (z1 + β) = bjky1z1 i+j+k=r j+k=r for some bjk ∈ k. Thus, the equality y z X y z L(1, , ) = ( − α)j( − β)k x x x x j+k=r implies V (y−αx, z−βx) ⊂ N. We conclude that τ(q) ≤ 2 if νq1 (S1) = νq(S).

Suppose that τ(q) = 2 and νq1 (S1) = νq(S). Then we must have α = β = 0, so that q1 ∈ N1. We further have

f1 = L(y1, z1) + x1Ω, so that τ(q1) ≥ τ(q). If τ(q) = 1, then L = czr for some (non-zero) c ∈ K, and we must have β = 0, so that q1 ∈ N1. We further have

f1 = L(z1) + x1Ω, so that τ(q1) ≥ τ(q). There is a similar analysis in cases 2 and 3.

Lemma 7.5. Suppose that C ⊂ Singr(S) is a non-singular curve and q ∈ C −1 is a closed point. Let π : V1 → V be the blow-up of C, E = π (C) the exceptional divisor, and S1 the strict transform of S on V1. Suppose that 108 7. Resolution of Surfaces in Positive Characteristic

(x, y, z) are regular parameters in OV,q (or OˆV,q) such that x = y = 0 are local equations of C at q, and x = 0 or x = y = 0 are local equations of a (possible formal) approximate manifold N of S at q. Let N1 be the strict −1 transform of N on V1. Suppose that q1 ∈ π (q). Then νq1 (S1) ≤ r, and if

νq1 (S1) = r, then τ(q) ≤ τ(q1) and q1 ∈ N1 ∩ E. Furthermore, N1 ∩ E = ∅ if τ(q) = 2, and N1 ∩ E is a curve which maps isomorphically onto spec(OC,q) (or spec(OˆC,q) in the case where N is formal) if τ(q) = 1.

We omit the proof of Lemma 7.5, as it is similar to Lemma 7.4.

Theorem 7.6. There exists a sequence of blow-ups of points in Singr(Si),

Vn → Vn−1 → · · · → V, where Si is the strict transform of S on Vi, so that all curves in Singr(Sn) are non-singular.

Proof. This is possible by Corollary 4.4 and since (by Lemma 7.4) the blow-up of a point in Singr(Si) can introduce at most one new curve into Singr(Si+1), which must be non-singular.

Theorem 7.7. Suppose that all curves in Singr(S) are non-singular, and

· · · → Vn → Vn−1 → · · · → V is a sequence of blow-ups of non-singular curves in Singr(Si) where Si is the strict transform of S on Vi. Then Singr(Si) is a union of non-singular curves and a finite number of points for all i. Further, this sequence is finite. That is, there exists an n such that Singr(Sn) is a finite set.

Proof. If π : V1 → V is the blow-up of the non-singular curve C in Singr(S), −1 and C1 ⊂ Singr(S1) ∩ π (C) is a curve, then by Lemma 7.5, C1 is non- singular and maps isomorphically onto C. Furthermore, the strict transform of a non-singular curve must be non-singular, so Singr(Si) must be a union of non-singular curves and a ﬁnite number of points for all i. Suppose that we can construct an inﬁnite sequence

· · · → Vn → Vn−1 → · · · → V by blowing up non-singular curves in Singr(Si). Then there exist curves Ci ⊂ Singr(Vi) for all i such that Ci+1 → Ci are dominant morphisms for all i, and ν (I ) = r OVi,Ci Si,Ci for all i. We have an inﬁnite sequence

(7.1) OV,C → OV1,C1 → · · · → OVn,Cn → · · · 7.2. τ(q) = 2 109 of local rings, where, after possibly reindexing to remove the maps which are the identity, each homomorphism is a localization of the blow-up of the maximal ideal of a 2-dimensional regular local ring. As explained in the characteristic zero proof of resolution of surface singularities (Lemma 5.10), we can view the OSi,Ci as local rings of a closed point of a curve on a non- singular surface deﬁned over a ﬁeld of transcendence degree 1 over K. By Theorem 3.15 (or Corollary 4.4), ν (I ) < r OVn,Cn Sn,Cn for large n, a contradiction. We can now state the main resolution theorem.

Theorem 7.8. Suppose that Singr(S) is a ﬁnite set. Consider a sequence of blow-ups

(7.2) · · · → Vn → · · · → V2 → V1 → V, where Vi+1 → Vi is the blow-up of a curve in Singr(Si) if such a curve exists, and Vi+1 → Vi is the blow-up of a point in Singr(Si) otherwise. Let Si be the strict transform of S on Vi. Then Singr(Si) is a union of non-singular curves and a ﬁnite number of points for all i. Further, this sequence is ﬁnite. That is, there exists Vn such that Singr(Sn) = ∅. Theorem 7.8 is an immediate consequence of Theorem 7.9 below, and Theorems 7.6 and 7.7. Theorem 7.9. Let the assumptions be as in the statement of Theorem 7.8, and suppose that q ∈ Singr(S) is a closed point. Then there exists an n such that all closed points qn ∈ Sn such that qn maps to q and νqn (Sn) = r satisfy τ(qn) > τ(q).

The proof of this theorem is very simple if τ(q) = 3. By Lemma 7.4 the multiplicity must drop at all points above q in the blow-up of q. We will prove Theorem 7.9 in the case τ(q) = 2 in Section 7.2 and in the case τ(q) = 1 in Section 7.3.

7.2. τ(q) = 2

In this section we prove Theorem 7.9 in the case that τ(q) = 2. Suppose that τ(q) = 2, and that there exists a sequence (7.2) of in- ﬁnite length. We can then choose an inﬁnite sequence of closed points qi ∈ Singr(Si) such that qi+1 maps to qi for all i. By Lemma 7.4, we have τ(qi) = 2 for all i, and each qi is isolated in Singr(Si). Thus each map Vi+1 → Vi is either the blow-up of the unique point qi ∈ Singr(Si) which maps to q, or it is an isomorphism over q. After 110 7. Resolution of Surfaces in Positive Characteristic

reindexing, we may assume that each map is the blow-up of a point qi such ˆ that qi+1 maps to qi for all i. Let Ri = OVi,qi . We have a sequence

(7.3) R0 → R1 → · · · → Ri → · · · of infinite length, where each Ri+1 is the completion of a local ring of a closed point of the blow-up of the maximal ideal of spec(Ri), and νqi (Si) = r and τ(qi) = 2 for all i. We will derive a contradiction. Since τ(q) = 2, there exist regular parameters (x, y, z) in R0 such that a local equation f = 0 of S in R0 has the form X i j k f = aijkx y z i+j+k≥r and the leading form of f is X i j L(x, y) = aij0x y . i+j=r For an arbitrary series X i j k g = bijkx y z ∈ R0, define k 1 γ (g) = min{ | b 6= 0 and i + j < r} ∈ ∪ {∞} xyz r − (i + j) ijk r!N with the convention that γxyz(g) = ∞ if and only if bijk = 0 whenever i + j < r. We have γxyz(g) < 1 if and only if νR0 (g) < r. Furthermore, r γxyz(g) = ∞ if and only if g ∈ (x, y) . Set γ = γxyz(g). Define X i j k (7.4) [g]xyz = bijkx y z . (i+j)γ+k=rγ There is an expansion X i j k (7.5) g = [g]xyz + bijkx y z . (i+j)γ+k>rγ

Returning to our series f, which is a local equation of S, set γ = γxyz(f), and k T = {(i, j) | = γ, i + j < r, for some k such that a 6= 0}. γ r − (i + j) ijk Then X i j γ(r−i−j) (7.6) [f]xyz = L + ai,j,γ(r−i−j)x y z .

(i,j)∈Tγ

Deﬁnition 7.10. [f]xyz is solvable if γ ∈ N and there exists α, β ∈ K such that γ γ [f]xyz = L(x − αz , y − βz ). 7.2. τ(q) = 2 111

Lemma 7.11. There exists a change of variables n n X i X i x1 = x − αiz , y1 = y − βiz i=1 i=1 such that [f]xyz is not solvable. Remark 7.12. We can always choose (x, y, z) that are regular parameters in OV0,q0 . Then (x, y, z) are also regular parameters in OV0,q0 . However, since we are only blowing up maximal ideals in (7.2), we may work with any formal system of regular parameters (x, y, z) in R0 (see Lemma 5.3).

Proof. We prove 7.11. If there doesn’t exist such a change of variables, then there exists an inﬁnite sequence of changes of variables for n ∈ N n n X γi X γi xn = x − αiz , yn − βiz i=1 i=1 such that γn ∈ N for all n, γn+1 = γxnynz(f), and γn+1 > γn for all n. Let ∞ ∞ X γi X γi x∞ = x − αiz , y∞ = y − βiz . i=1 i=1 Thus

γ∞ = γx∞,y∞,z(f) = ∞ and ∞ ∞ X i X i r f ∈ (x − αiz , y − βiz ) ⊂ R0. i=1 i−1

But by construction of V0, p is isolated in Singr(S). If I ⊂ OV,q is the reduced ˆ ideal deﬁning Singr(S) at q, then by Remark A.21, I = IR0 is the reduced ideal whose support is the locus where f has multiplicity r in spec(R0). ˆ P i P i Thus (x, y, z) = I ⊂ (x − αiz , y − βiz ), which is impossible.

We may thus assume that [f]xyz is not solvable. Since the leading form of f is L(x, y) and τ(p) = 2, R1 has regular parameters (x1, y1, z1) deﬁned by

(7.7) x = x1z1, y = y1z1, z = z1. f Let f1 = zr = 0 be a local equation of the strict transform of f in R1. Lemma 7.13.

1. γx1y1z(f1) = γxyz(f) − 1. 1 2. [f1]x1y1z = zr [f]xyz, so [f]xyz not solvable implies [f1]xyz not solvable. 1 3. If γx1y1z(f1) > 1, then the leading form of f1 is L(x1, y1) = zr L(x, y). 112 7. Resolution of Surfaces in Positive Characteristic

Proof. The lemma follows from substitution of (7.7) in (7.4), (7.5) and (7.6).

We claim that the case γx1,y1,z(f1) = 1 cannot occur (with our assumption that νq1 (S1) = r and τ(q1) = 2). Suppose that it does. Then the leading form of f1 is [f1]x1y1z, and there exist a form Ψ of degree r and a, b, c, d, e, f ∈ K such that

[f1]x1y1z = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz). We have an expression

L(x1, y1) + zΩ = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz) so that L(x, y) = Ψ(ax + by, dx + ey).

We have ae−bd 6= 0, since τ(q) = 2. By Lemma 7.14 below, [f1]x1y1z is thus solvable, a contradiction to 2 of Lemma 7.13.

Since γx1y1z(f1) < 1 implies νR1 (f1) < r, it now follows from Lemma 7.13 that after a ﬁnite sequence of blow-ups R0 → Ri, where Ri has regular parameters (xi, yi, zi) with i i x = xizi, y = yizi, z = zi.

By 3 of Lemma 7.13, we reach a reduction νqi (Si) < r or νqi (Si) = r, τ(qi) = 3, a contradiction to our assumption that (7.3) has inﬁnite length. Thus Theorem 7.9 has been proven when τ(q) = 2.

Lemma 7.14. Let Φ = Φ(x, y), Ψ = Ψ(u, v) be forms of degree r over K and suppose that Φ(x, y) = Ψ(ax + by, dx + ey) for some a, b, d, e ∈ K with ae − bd 6= 0. Then for all c, f ∈ K there exist α, β ∈ K such that Φ(x + αz, y + βz) = Ψ(ax + by + cz, dx + ey + fz).

Proof. Indeed, Φ(x + αz, y + βz) = Ψ(a(x + αz) + b(y + βz), d(x + αz) + e(y + βz)) = Ψ(ax + by + (aα + bβ)z, dx + ey + (dα + eβ)z), so we need only solve aα + bβ = c, dα + eβ = f for α, β. 7.3. τ(q) = 1 113

7.3. τ(q) = 1

In this section we prove Theorem 7.9 in the case that τ(q) = 1. We now assume that τ(q) = 1. Suppose that there is a sequence (7.2) which does not terminate. We can then choose a sequence of points qn ∈ Vn, with q0 = q, such that qn+1 maps to qn for all n and νqn (Sn) = r, τ(qn) = 1 for all n. After possibly reindexing, we may assume that for all n, qn is on ˆ the subvariety of Vn which is blown up under Vn+1 → Vn. Let Rn = OVn,qn for n ≥ 0. We then have an inﬁnite sequence

(7.8) R = R0 → R1 → · · · → Rn → · · · .

Let Ji ⊂ OVi,qi be the reduced ideal deﬁning Singr(Si) at qi. Then by

Lemmas 7.4 and 7.5, Ji is either the maximal ideal mi of OVi,qi or a regular ˆ height 2 prime ideal pi in OVi,qi . Then the multiplicity r locus of OSi,qi (which is a quotient of Ri by a principal ideal) is defined by the reduced ideal Jî = JiRi by Remark A.21. We have Jî =m ˆ i = miRi or Jî =p î = piRi, a regular prime in Ri. By the construction of the sequence (7.2), we see that

OVi+1,qi+1 is a local ring of the blow-up of mi if Ji = mi, and a local ring of the blow-up of pi otherwise. Thus Ri+1 is the completion of a local ring of the blow-up ofm ˆ i orp î. We are thus free to work with formal parameters and equations (which ˆ define the ideal ISi,qi = ISi,qi Ri) in the Ri, since the idealsm ˆ i andp î are ˆ determined in Ri by the multiplicy r locus of Singr(OSi,qi ). Suppose that T = K[[x, y, z]] is a power series ring, r ∈ N and X i j k g = bijkx y z ∈ T. We can construct a polygon in the following way. Define i j ∆ = ∆(g; x, y, z) = {( , ) ∈ 2 | k < r and b 6= 0}. r − k r − k Q ijk 2 Let |∆| be the smallest convex set in R such that ∆ ⊂ |∆| and (a, b) ∈ |∆| implies (a + c, b + d) ∈ |∆| for all c, d ≥ 0. For a ∈ R, let S(a) be the line through (a, 0) with slope −1, and V (a) the vertical line through (a, 0).

Suppose that |∆|= 6 ∅. We deﬁne αxyz(g) to be the smallest a appearing in any (a, b) ∈ |∆|, and βxyz(g) to be the smallest b such that

(αxyz(g), b) ∈ |∆|.

Let γxyz(g) be the smallest number γ such that S(γ)∩|∆|= 6 ∅ and let δxyz(g) be such that (γxyz(g)−δxyz(g), δxyz(g)) is the lowest point on S(γxyz(g))∩|∆|. Then (αxyz(g), βxyz(g)) and (γxyz(g) − δxyz(g), δxyz(g)) are vertices of |∆|. 114 7. Resolution of Surfaces in Positive Characteristic

Deﬁne xyz(g) to be the absolute value of the largest slope of a line through (αxyz(g), βxyz(g)) such that no points of |∆(g; x, y, z)| lie below it. Lemma 7.15. 1 1 1. The vertices of |∆| are points of ∆, which lie on the lattice r! Z× r! Z. 2. νR(g) < r holds if and only if |∆| contains a point on S(c) with c < 1, which in turn holds if and only if there is a vertex (a, b) with a + b < 1. r 3. αxyz(g) < 1 if and only if g 6∈ (x, z) . 4. A vertex of |∆| lies below the line b = 1 if and only if g 6∈ (y, z)r.

Proof. Lemma 7.15 follows directly from the deﬁnitions.

Suppose that g ∈ T is such that νT (g) = r. Let X i j k L = bijkx y z i+j+k=r be the leading form of g.(x, y, z) will be called good parameters for g if b00r = 1.

We now suppose that νT (g) = r, and (x, y, z) are good parameters for g. Let ∆ = ∆(g; x, y, z) and suppose that (a, b) is a vertex of |∆|. Let i j S = {k | ( , ) = (a, b) and b 6= 0}. (a,b) r − k r − k ijk Deﬁne ab r X a(r−k) b(r−k) k {g}xyz = z + ba(r−k),b(r−k),kx y z .

k∈S(a,b) ab We will say that {g}xyz is solvable (or that the vertex (a, b) is not prepared on |∆(g; x, y, z)|) if a, b are integers and ab a b r {g}xyz = (z − ηx y ) for some η ∈ K. We will say that the vertex (a, b) is prepared on ab ab |∆(g; x, y, z)| if {g}x,y,z is not solvable. If {g}xyz is solvable, we can make an (a, b) preparation, which is the change of parameters a b z1 = z − ηx y . If all vertices (a, b) of |∆| are prepared, then we say that (g; x, y, z) is well prepared. Lemma 7.16. Suppose that T is a power series ring over K, g ∈ T is such that νT (g) = r, τ(g) = 1, (x, y, z) are good parameters of g, and (g; x, y, z) is well prepared. Then z = 0 is an approximate manifold of g = 0. 7.3. τ(q) = 1 115

Proof. νT (g) = r implies all vertices (a, b) of |∆(g; x, y, z)| satisfy a+b ≥ 1. The leading form of g is X i j k L = bijkx y z , i+j+k=r where b00r = 1. τ(g) = 1 implies there exists a linear form ax + by + cz such that (ax + by + cz)r = L.

As b00r = 1, we must have c = 1. If a 6= 0, then (1, 0) is a vertex of |∆(g; x, y, z)|, which is not prepared since (1,0) r {g}xyz = (z + ax) . Thus a = 0. If b 6= 0, then (0, 1) is a vertex of |∆(g; x, y, z)|, which is not prepared since (0,1) r {g}xyz = (z + by) . r Thus L = z , and z = 0 is an approximate manifold of g = 0. Lemma 7.17. Consider the terms in the expansion k X k (7.9) h = ηk−λ xi+(k−λ)ayj+(k−λ)bzλ λ 1 λ=0 a b i j k obtained by substituting z1 = z − ηx y into the monomial x y z . Define a 3 projection for (a, b, c) ∈ N such that c < r and a b π(a, b, c) = ( , ). r − c r − c Then: 1. Suppose that k < r. Then the exponents of monomials in (7.9) with non-zero coefficients project into the line segment joining (a, b) to i j ( r−k , r−k ). i j a. If (a, b) = ( r−k , r−k ), then all these monomials project to (a, b). i j i j k b. If (a, b) 6= ( r−k , r−k ), then x y z1 is the unique monomial in i j (7.9) which projects onto ( r−k , r−k ). No monomial in (7.9) projects to (a, b). 2. Suppose that r ≤ k, and (i, j, k) 6= (0, 0, r). Then all exponents in (7.9) with non-zero coefficients and z1 exponent less that r project into 2 (a, b) + Q≥0 − {(a, b)}. 3. Suppose that (i, j, k) = (0, 0, r). Then all exponents in (7.9) with non-zero coefficients and z1 exponent less than r project to (a, b). 116 7. Resolution of Surfaces in Positive Characteristic

Proof. If k 6= r, we have i+(k−λ)a (r−k) i r−λ = a + (r−λ) r−k − a , j+(k−λ)b (r−k) j r−λ = b + (r−λ) r−k − b . r−k Suppose that k < r. Since 0 ≤ λ ≤ k, we have 0 < r−λ ≤ 1. Thus i + (k − λ)a j + (k − λ)b , r − λ r − λ i j is on the line segment joining (a, b) to ( r−k , r−k ), and 1 follows. r−k i j If k > r and 0 ≤ λ < r ≤ k, then r−λ < 0, r−k − a < 0 and r−k − b < 0. Thus 2 follows if r 6= k. Suppose that k = r and 0 ≤ λ < r. Then i + (k − λ)a j + (k − λ)b i j , = + a, + b , r − λ r − λ r − λ r − λ and the last case of 2 and 3 follow. We deduce from this lemma that a b Lemma 7.18. Suppose that z1 = z − ηx y is an (a, b) preparation. Then:

1. |∆(g; x, y, z1)| ⊂ |∆(g; x, y, z)| − {(a, b)}. 2. If (a0, b0) is another vertex of |∆(g; x, y, z)|, then (a0, b0) is a vertex a0,b0 a0,b0 of |∆(g; x, y, z1)| and {g}x,y,z1 is obtained from {g}x,y,z by substituting z1 for z. Example 7.19. It is not always possible to well prepare after a ﬁnite number of vertex preparations. Consider over a ﬁeld of characteristic 0 (or p > r) m m+1 r gm = y(y − x)(y − 2x) ··· (y − rx) + (z − x + x − · · · ) 1 1 with m ≥ 2. The vertices of |∆(g; x, y, z)| are (0, 1 + r ), (1, r ) and (m, 0). m We can remove the vertex (m, 0) by the (m, 0) preparation z1 = z − x . Thus x2 r g = y(y − x)(y − 2x) ··· (y − rx) + (z − 1+x ) = y(y − x)(y − 2x) ··· (y − rx) + (z − x2 + x3 − · · · )r can only be well prepared by the formal substitution ∞ X i i z1 = z − (−1) x . i=2

Lemma 7.20. Suppose that g is reduced, νT (g) = r, τ(g) = 1 and (x, y, z) are good parameters for g. Then there is a formal series φ(x, y) ∈ K[[x, y]] such that under the substitution z = z1 + φ(x, y), (x, y, z1) are good parameters for g and (g; x, y, z1) is well prepared. 7.3. τ(q) = 1 117

Proof. Let vL be the lowest vertex of |∆(g; x, y, z)|. Let h(vL) be the second coordinate of vL. Set b = h(vL). If vL is not prepared, make a vL preparation a b z1 = z − ηx y (where (a, b) = vL) to remove vL in |∆(g; x, y, z1)|.

Let vL1 be the lowest vertex of |∆(g; x, y, z1)|. If vL1 is not prepared and h(vL1 ) = h(vL), we can again prepare vL1 by a vL1 preparation z2 = a b z1 − η1x 1 y . We can iterate this procedure to either achieve |∆(g; x, y, zn)| such that the lowest vertex vLn is solvable or h(vLn ) > h(vL), or we can construct an inﬁnite sequence of vLn preparations (with h(vLn ) = b for all n)

an b zn+1 = zn − ηnx y , where (an, b) = vLn , such that the lowest vertex vLn of |∆(g; x, y, zn)| is not prepared and h(vLn ) = h(vL) for all n. Since an+1 > an for all n, 0 P∞ ai b we can then make the formal substitution z = z − i=0 ηix y , to get 0 |∆(g; x, y, z )| whose lowest vertex vL0 satisﬁes h(vL0 ) > h(vL). In summary, there exists a series Φ0(x) such that if we set z0 = z − h(v ) 0 0 y L Φ (x), and vL0 is the lowest vertex of |∆(g; x, y, z )|, then either vL0 is prepared or h(vL0 ) > h(vL). By iterating this procedure, we construct a series Φ(x, y) such that if z = z −Φ(x, y), then either the lowest vertex vL of |∆(g; x, y, z)| is prepared, or |∆(g; x, y, z)| = ∅. This last case only occurs if g = unit zr, and thus cannot occur since g is assumed to be reduced.

We can thus assume that vL is prepared. We can now apply the same procedure to vT , the highest vertex of |∆(g; x, y, z)|, to reduce to the case where vT and vL are both prepared. Then after a ﬁnite number of preparations we ﬁnd a change of variables z0 = z − Φ(x, y) such that (g; x, y, z0) is well prepared.

We will also consider change of variables of the form

n y1 = y − ηx for η ∈ K, n a positive integer, which we will call translations.

Lemma 7.21. Consider the expansion

j X j (7.10) h = ηj−λ xi+(j−λ)nyλzk λ 1 λ=0 n i j k obtained by substituting y1 = y − ηx into the monomial x y z . Consider 3 the projection for (a, b, c) ∈ N with c < r deﬁned by a b π(a, b, c) = ( , ). r − c r − c 118 7. Resolution of Surfaces in Positive Characteristic

i j i j k Suppose that k < r. Set (a, b) = ( r−k , r−k ). Then x y1z is the unique monomial in (7.10) whose exponents project onto (a, b). All other monomials in (7.10) with non-zero coeﬃcient project to points below (a, b) on the line 1 through (a, b) with slope − n . Proof. The slope of the line through the points (a, b) and i + (j − λ)n λ , r − k r − k is j − λ 1 = − . i − (i + (j − λ)n) n λ j Since r−k < r−k for λ < j, the lemma follows.

Deﬁnition 7.22. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1 and (x, y, z) are good parameters for g. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), = xyz(g). Then (g; x, y, z) will be said to be very well prepared if it is well prepared and one of the following conditions holds.

1. (γ − δ, δ) 6= (α, β), and if we make a translation y1 = y − ηx, with subsequent well preparation z1 = z − Φ(x, y), then

αxy1z1 (g) = α, βxy1z1 (g) = β, γxy1z1 (g) = γ and

δxy1z1 (g) ≤ δ. 2. (γ − δ, δ) = (α, β), and one of the following cases holds: a. = 0. 1 b. 6= 0 and is not an integer. 1 c. 6= 0 and n = is a (positive) integer and for any η ∈ K, if n y1 = y − ηx is a translation, with subsequent well prepara-

tion z1 = z − Φ(x, y), then xy1z(g) = . Further, if (c, d) is the lowest point on the line through (α, β) with slope − in |∆(g; x, y, z1)| and (c1, d1) is the lowest point on this line in |∆(g; x, y1, z1)|, then d1 ≤ d.

Lemma 7.23. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1 and (x, y, z) are good parameters for g. Then there are formal substitutions

z1 = z − φ(x, y), y1 = y − ψ(x), where φ(x, y), ψ(x) are series such that (g; x, y1, z1) is very well prepared.

Proof. By Lemma 7.20, we may suppose that (g; x, y, z) is well prepared. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), = xyz(g). Let (γ − t, t) be the highest vertex on |∆(g; x, y, z)| on the line a + b = γ. Let 7.3. τ(q) = 1 119

η ∈ K and consider the translation y1 = y − ηx. By Lemma 7.21, (γ − t, t) and (α, β) are prepared vertices of |∆(g; x, y1, z)|. Thus if z1 = z − Φ(x, y) is a well preparation of (g; x, y1, z), then (γ − t, t) and (α, β) are vertices of |∆(g; x, y1, z1)|.

We can thus choose η ∈ K so that after the translation y1 = y − ηx, and a subsequent well preparation z1 = z − Φ(x, y), we have αxy1z1 = α,

βxy1z1 = β, γxy1z1 = γ and δxy1z1 ≥ δ is a maximum for substitutions y1 = y − ηx, z1 = z − Φ(x, y) as above. If (γ − δxy1z1 , δxy1z1 ) 6= (α, β), then after replacing (x, y, z) with (x, y1, z1), we have achieved condition 1 of the lemma, so that (g; x, y, z) is very well prepared. We now assume that (g; x, y, z) is well prepared, and (α, β) = (γ − δ, δ). 1 If = 0 or is not an integer, then (g; x, y, z) is very well prepared. 1 Suppose that n = is an integer. We then choose η ∈ K such that with n the translation y1 = y − ηx , and subsequent well preparation, we maximize d for points (c, d) of the line through (α, β) with slope − on the boundary of |∆(f; x, y1, z1)|. By Lemma 7.21 α, β and γ are not changed. If we now have d 6= β, we are very well prepared. If this process does not end after a ﬁnite number of iterations, then there exists an inﬁnite sequence of changes of variables (translations followed by well preparations)

ni+1 yi = yi+1 + ηi+1x , zi = zi+1 + φi+1(x, yi+1) such that ni+1 > ni for all i. Given n ∈ N, there exists σ(n) ∈ N such that σ(n) > σ(n − 1) for all n, and i ≥ σ(n) implies all vertices (a, b) of |∆(g; x, yi, zi)| below (α, β) have a ≥ n. This last condition follows since all 1 1 vertices must lie in the lattice r! Z × r! Z, and there are only ﬁnitely many points common to this lattice and the region 0 ≤ a ≤ n, 0 ≤ b ≤ β. Thus n x | φi(x, yi) if i ≥ σ(n). Let

i j X X nk Ψi(x, y) = φj(x, y − ηkx ), j=1 k=1 so that zi = z − Ψi(x, y) for all i. {Ψi(x, y)} is a Cauchy sequence in 0 P∞ K[[x, y]]. Thus z = z − i=1 φi(x, yi) is a well deﬁned series in K[[x, y, z]]. 0 P∞ ni 0 0 Set y = y − i=1 ηix . Then |∆(g; x, y , z )| has the single vertex (α, β), 0 0 and (g; x, y , z ) is thus very well prepared.

Deﬁnition 7.24. Suppose that g ∈ T , g is reduced, νT (g) = r, τ(g) = 1, and (x, y, z) are good parameters for g. We consider 4 types of monodial transforms T → T1, where T1 is the completion of the local ring of a monoidal transform of T , and T1 has regular parameters (x1, y1, z1) related to the regular parameters (x, y, z) of T by one of the following rules. 120 7. Resolution of Surfaces in Positive Characteristic

T1. Singr(g) = V (x, y, z),

x = x1, y = x1(y1 + η), z = x1z1, g with η ∈ K. Then g1 = r is the strict transform of g in T1, and x1

if νT1 (g1) = r and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

T2. Singr(g) = V (x, y, z),

x = x1y1, y = y1, z = y1z1. g Then g1 = r is the strict transform of g in T1, and if νT1 (g1) = r y1 and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

T3. Singr(g) = V (x, z),

x = x1, y = y1, z = x1z1. g Then g1 = r is the strict transform of g in T1, and if νT1 (g1) = r x1 and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

T4. Singr(g) = V (y, z),

x = x1, y = y1, z = y1z1. g Then g1 = r is the strict transform of g in T1, and if νT1 (g1) = r y1 and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1. Lemma 7.25. With the assumptions of Deﬁnition 7.24, suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1, (x, y, z) are good parameters for g, (x, y, z) and (x1, y1, z1) are related by a transformation of one of the above types

T1-T4, and νT1 (g) = r1, τ(g1) = 1. Then there is a 1-1 correspondence

σ : ∆(g, x, y, z) → ∆(g1, x1, y1, z1) deﬁned by 1. σ(a, b) = (a + b − 1, b) if the transformation is a T1 with η = 0, 2. σ(a, b) = (a, a + b − 1) if the transformation is a T2, 3. σ(a, b) = (a − 1, b) if the transformation is a T3, 4. σ(a, b) = (a, b − 1) if the transformation is a T4.

The proof of Lemma 7.25 is straightforward, and is left to the reader. Lemma 7.26. In each of the four cases of the preceeding lemma, if σ(a, b) = (a1, b1) is a vertex of |∆(g1; x1, y1, z1)|, then (a, b) is a vertex of |∆(g; x, y, z)|, and if (g; x, y, z) is (a, b) prepared then (g1; x1, y1, z1) is (a1, b1) prepared. In particular, (g1; x1, y1, z1) is well prepared if (g; x, y, z) is well prepared. 7.3. τ(q) = 1 121

To see that (g1; x1, y1, z1) is (a1, b1) prepared if (g; x, y, z) is (a, b) prepared, we observe that  1 {g}a,b if we perform a T3 transformation  xr x,y,z a1,b1 or a T1 transformation with η = 0, {g1}x1,y1,z1 =  1 a,b  yr {g}x,y,z if we perform a T2 or T4 transformation. Lemma 7.27. Suppose that assumptions are as in Deﬁnition 7.24 and (x, y, z) and (x1, y1, z1) are related by a T3 transformation. Suppose that

νT1 (g1) = r and τ(g1) = 1. If (g; x, y, z) is very well prepared, then (g1; x1, y1, z1) is very well prepared. Moreover,

βx1y1z1 (g1) = βxyz(g), δx1y1z1 (g1) = δxyz(g), x1y1z1 (g1) = xyz(g) and

αx1y1z1 (g1) = αxyz(g) − 1, γx1y1z1 (g1) = γxyz(g) − 1.

We further have Singr(g1) ⊂ V (x1, z1).

Proof. Well preparation is preserved by Lemma 7.26. Suppose (g; x, y, z) is very well prepared and

(αxyz, βxyz) 6= (γxyz − δxyz, δxyz).

Let γ = γxyz(g), γ1 = γxyz(g1) = γ − 1. The terms in g which contribute to the line S(γ) are

X i j k (7.11) bijkx y z . i + j + γk = rγ k < r They are transformed to

X i+k−r j k (7.12) bijkx1 y1z1 , (i + k − r) + j + γ1k = rγ1 k < r 122 7. Resolution of Surfaces in Positive Characteristic

which are the terms in g1 which contribute to the line S(γ1). We have j δ (g) = δ (g ) = min{ | b 6= 0 in (7.11)}. xyz x1,y1,z1 1 r − k ijk If a translation y = y + ηx transforms (7.11) into

X i j k (7.13) cijkx y z , i + j + γk = rγ k < r then the translation y1 = y1 + ηx transforms (7.12) into X i j k (7.14) cijkx y1z1 . (i + k − r) + j + γ1k = rγ1 k < r

Since (7.13) and (7.14) are the terms which contribute to δxyz(g) and

δx1y1z1 (g1), we have δxyz(g) = δx1y1z1 (g1) and (g; x, y, z) well prepared implies (g1; x1, y1, z1) well prepared. We can make a similiar argument if

(αxyz, βxyz) = (γxyz − δxyz, δxyz).

The statement Singr(g1) ⊂ V (x1, z1) follows since Singr(g) = V (x, y, z), so that Singr(g1) ⊂ V (x1), the exceptional divisor of T3, and Singr(g1) ⊂ V (z1) since z = 0 is an approximate manifold of g = 0 (Lemmas 7.16 and 7.5). Lemma 7.28. Suppose that assumptions are as in Deﬁnition 7.24, (x, y, z) and (x1, y1, z1) are related by a T1 transformation with η = 0, (g; x, y, z) is 0 0 very well prepared and νT1 (g1) = r, τ(g1) = 1. Let (g1; x1, y , z ) be a very well preparation of (g1; x1, y1, z1). Let

α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), = xyz(g),

0 0 0 0 0 0 α1 = αx1y z (g1), β1 = βx1y z (g1), γ1 = γx1y z (g1),

0 0 0 0 δ1 = δx1y z (g1), 1 = x1y z (g1). Then 1 1 (β1, δ1, , α1) < (β, δ, , α) 1 1 in the lexicographical order, with the convention that if = 0, then = ∞. If β = β, δ = δ and 6= 0, then 1 = 1 − 1. 1 1 1 0 We further have Singr(g1) ⊂ V (x1, z ). 7.3. τ(q) = 1 123

Proof. Let σ : ∆(g; x, y, z) → ∆(g1; x1, y1, z1) be the 1-1 correspondence of Lemma 7.25 1, which is defined by σ(a, b) = (a+b−1, b). σ transforms lines m of slope m 6= −1 to lines of slope m+1 , and transforms lines of slope −1 to vertical lines. First suppose that − ≤ −1. Then (α, β) 6= (γ −δ, δ), and we are in case 1 of Definition 7.22. Then the line segment through (α, β) and (γ − δ, δ) (which has slope ≤ −1) is transformed to a segment with positive slope, or a vertical line. Thus βx1,y1,z1 = δ < β. Since (g; x1, y1, z1) is well prepared, after very well preparation we have β1 < β. 1 Suppose that −1 < − ≤ − 2 . We have (α, β) = (γ − δ, δ). Then the line segment through (α, β) and (c, d) (with the notation of Definition 7.22) is transformed to the line segment through (α + β − 1, β) and (c + d − 1, d), which has slope − − = < −1. x1y1z1 − + 1 Thus

(αx1y1z1 , βx1y1z1 ) = (α + β − 1, β) and

(γx1y1z1 (g1) − δx1y1z1 (g1), δx1y1z1 (g1)) 6= (αx1y1z1 (g1), βx1y1z1 (g1)).

Let (γx1y1z1 (g1) − t, t) be the highest point on the line S(γx1y1z1 (g1)) in |∆(g1; x1, y1, z1)|. By Lemma 7.26, (g1; x1, y1, z1) is well prepared. Thus (by Lemma 7.21) very well preparation does not aﬀect the vertices

(αx1y1z1 (g1), βx1y1z1 (g1)) and (γx1y1z1 (g1) − t, t) of |∆(g1; x1, y1, z1)| which are below the horizontal line of points whose second coordinate is d. Thus β1 = β and δ1 ≤ t ≤ d < δ. 1 Suppose that − 2 < − < 0. In this case we have (α, β) = (γ−δ, δ). Then the line segment through (α, β) and (c, d) (with the notation of Deﬁnition 7.22) is transformed to the line segment through (α+β−1, β) and (c+d−1, d) which has slope − − = , x1y1z1 − + 1 so that −1 < −x1y1z1 < 0. Thus

(αx1y1z1 , βx1y1z1 ) = (γx1y1z1 − δx1y1z1 , δx1y1z1 ) = (α + β − 1, β), and (c1, d) = (c + d − 1, d) is the lowest point on the line through the point (αx1y1z1 , βx1y1z1 ) with slope −x1y1z1 which is on |∆(g1; x1, y1, z1)|. By Lemma 7.26, (g1; x1, y1, z1) is well prepared. We will now show that 1 (g1; x1, y1, z1) is very well prepared. If 6∈ N, then this is immediate, x1,y1,z1 so suppose that 1 n = ∈ N. x1y1z1 124 7. Resolution of Surfaces in Positive Characteristic

1 We necessarily have n > 1, and = n + 1. If we can remove the ver- n tex (c1, d) from |∆(g1; x1, y1, z1)| by a translation y1 = y1 − ηx1 , then the translation y = y − ηxn+1 will remove the vertex (c, d) from |∆(g; x, y, z)|, a contradiction to the assumption that (g; x, y, z) is very well prepared.

Thus (g1; x1, y1, z1) is very well prepared, with β1 = β, δ1 = δ and 1 = 1 − 1. 1 The ﬁnal case is when = 0. Then β = δ,(g1; x1, y1, z1) is very well prepared,

(α1, β1) = (γ1 − δ1, δ1) = (α + β − 1, β) and = 0. Thus β = β, δ = δ and 1 = 1 = ∞. Also, α < α, since 1 1 1 1 1 Singr(g) = V (x, y, z) (by assumption), so = 0 implies β < 1.

By Lemmas 7.4 and 7.16, Singr(g1) ⊂ V (x1, z1). If Singr(g1) = V (x1, z1), then all vertices of |∆(g1; x1, y1, z1)| have the ﬁrst cordinate ≥ 1, and thus 0 Singr(g1) = V (x1, z ). Lemma 7.29. Suppose that assumptions are as in Deﬁnition 7.24, (x, y, z) and (x1, y1, z1) are related by a T2 or a T4 transformation, and (g; x, y, z) is well prepared. Suppose that νT1 (g1) = r and τ(g1) = 1. We have that 0 0 (g1; x1, y1, z1) is well prepared and Singr(g1) ⊂ V (y1, z1). Let (g1; x1, y , z ) be a very well preparation. Then

0 0 βx1,y1,z1 = βx1,y ,z (g1) < βxyz(g).

Proof. First suppose that T → T1 is a T2 transformation. The correspondence σ : ∆(g; x, y, z) → ∆(g1; x1, y1, z1) of Lemma 7.25 2. is deﬁned by σ(a, b) = (a, a + b − 1). σ takes lines of slope m to lines of slope m + 1. Thus

(αx1y1z1 (g1), βx1y1z1 (g1)) = (αxyz(g), αxyz(g) + βxyz(g) − 1).

Since Singr(g) = V (x, y, z) by assumption, we have αxyz(g) < 1, and thus

βx1y1z1 (g1) < βxyz(g).

(g1; x1, y1, z1) is well prepared by Lemma 7.26. Therefore, the vertex

(αx1y1z1 (g1), βx1y1z1 (g1))

0 0 is not aﬀected by very well preparation. We thus have βx1y z (g1) < βxyz(g). Now suppose that T → T1 is a T4 transformation. The 1-1 correspondence σ : ∆(g; x, y, z) → ∆(g1; x1, y1, z1) of Lemma 7.25 4 is deﬁned by

σ(a, b) = (a, b − 1). Thus βx1y1z1 (g1) < βxyz(g). Also, (g1; x1, y1, z1) is well prepared by Lemma 7.26, and the vertex (αx1y1z1 (g1), βx1y1z1 (g1)) is not 0 0 aﬀected by very well preparation. Thus βx1y z (g1) = βx1y1z1 < βxyz(g). Lemma 7.30. Suppose that p is a prime, s is a non–negative integer and s r0 is a positive integer such that p - r0. Let r = r0p . Then: 7.3. τ(q) = 1 125

r s 1. λ ≡ 0 mod p, if p - λ, 0 ≤ λ ≤ r is an integer. r r0 2. λps ≡ λ mod p, if 0 ≤ λ ≤ r0 is an integer.

r ps ps r Proof. Compare the expansions over Zp of (x + y) = (x + y ) 0 . Theorem 7.31. Suppose that assumptions are as in Deﬁnition 7.24, (x, y, z) and (x1, y1, z1) are related by a T1 transformation with η 6= 0, and (g; x, y, z) is very well prepared. Suppose that νT1 (g1) = r, τ(g1) = 1 and βxyz(g) > 0. 0 0 0 0 Then there exist good parameters (x, y1, z1) in T1 such that (g1; x1, y1, z1) is 0 very well prepared, with β 0 0 (g ) < β (g) and Sing (g ) ⊂ V (x , z ). x1,y1,z1 1 x,y,z r 1 1 1

Proof. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g). Lemma 7.16 implies z = 0 is an approximate manifold of g = 0. Thus (7.15) α + β > 1. Apply the translation y0 = y − ηx and well prepare by some substitution 0 0 0 z = z − Φ(x, y ). This does not change (α, β) or γ. Set δ = δx,y0,z0 (g). 0 First assume that δ < β. We have regular parameters (x1, y1, z1) in T1 0 0 such that x = x1, y = x1y1, z = x1z1 by Lemma 7.16, so we have a 1-1 correspondence (by Lemma 7.25 1)

0 0 σ : ∆(g; x, y , z ) → ∆(g1; x1, y1, z1) deﬁned by σ(a, b) = (a+b−1, b). (g1; x1, y1, z1) is well prepared (by Lemma 7.26). We are essentially in case 1 (− ≤ −1) of the proof of Lemma 7.28 0 0 (this case only requires well preparation), so that (α1, β1) = σ(γ − δ , δ ) 0 and β1 = δ < β. After very well preparation, we thus have regular pa- 0 0 0 0 rameters x1, y1, z1 in T1 such that (g1; x1, y1, z1) is very well prepared and β 0 0 (g ) < β (g). x1,y1,z1 1 xyz We have thus reduced the proof to showing that δ0 < β. If (α, β) 6= (γ − δ, δ), we have δ0 ≤ δ < β since (g; x, y, z) is very well prepared. P i j k Let g = aijkx y z . Suppose that (α, β) = (γ − δ, δ). Set i j W = {(i, j, k) ∈ 3 | k < r and ( , ) = (α, β)}, N r − k r − k

X i j k F = aijkx y z . (i,j,z)∈W By assumption, zr + F is not solvable. Moreover,

j X X j (7.16) F (x, y, z) = F (x, y0 + ηx, z) = a ηλ xi+λ(y0)j−λzk. ijk λ (i,j,k)∈W λ=0 126 7. Resolution of Surfaces in Positive Characteristic

By Lemma 7.21, the terms in the expansion of g(x, y0 + ηx, z) contributing to (γ, 0) in |∆(g; x, y0, z)|, where γ = α + β, are

X j i+j k F(γ,0) = aijkη x z 6= 0. (i,j,k)∈W

If (g; x, y0, z) is (γ, 0) prepared, then δ0 = 0 < β. Suppose that

γ0 r {g}x,y0,z = z + F(γ,0) is solvable. Then γ ∈ N, and there exists ψ ∈ K such that

γ r r (7.17) (z − ψx ) = z + F(γ,0),

−ψ so that, with ω = ηβ ∈ K, for 0 ≤ k < r, we have: r 1. If r−k 6= 0 (in K), then i = α(r − k), j = β(r − k) ∈ N and r (7.18) a = ωr−k. ijk r − k

r 2. If i = α(r − k), j = β(r − k) ∈ N and r−k = 0 (in K), then aijk = 0. s Thus (by Lemma 7.30) aijk = 0 if p - k. Suppose that K has characteristic 0. By Remark 7.32, we obtain a (γ,0) 0 contradiction to our assumption that {g}xy0z is solvable. Thus 0 = δ < β if K has characteristic 0. s Now we consider the case where K has characteristic p > 0, and r = p r0 s s with p - r0, r0 ≥ 1. By (7.18) and Lemma 7.30, we have i = αp , j = βp ∈ N s and ai,j,(r0−1)p 6= 0. s t We have an expression βp = ep , where p - e. Suppose that t ≥ s. Then β ∈ N, which implies α = γ − β ∈ N, so that (z + ωxαyβ)r = zr + F, a contradiction, since zr + F is by assumption not solvable. Thus t < s. Suppose that e = 1. Then β = pt−s < 1 and α < 1 (since we must have Singr(g) = V (x, y, z)) imply γ = α + β = 1, a contradiction to (7.15). Thus e > 1. Also,

s s s s zr + F = (zp + ωp xαp yβp )r0 = zps + ωps xαps (y0 + ηx)βps r0 s s s h t t t ier0 = zp + ωp xαp (y0)p + ηp xp . 7.3. τ(q) = 1 127

Now make the (γ, 0) preparation z = z0 − ηβωxγ (from (7.17)) so that (g; x, y0, z0) is (γ, 0) prepared. Let G = zr + F . Then

s s t t s t t r0 G = (z0)p + eωp ηp (e−1)(y0)p xαp +p (e−1) + (y0)2p Ω) 0 psr h ps pt(e−1) 0 pt αps+pt(e−1) 0 2pt i 0 ps(r −1) = (z ) 0 + r0 eω η (y ) x + (y ) Ω (z ) 0 s 0 0 p (r0−2) 0 +Λ2(x, y )(z ) + ··· + Λr0 (x, y )

0 0 ipt for some polynomials Ω(x, y ), Λ2,..., Λr0 , where (y ) | Λi for all i. Since all contributions to S(γ) ∩ |∆(g; x, y0, z0)| must come from this polynomial (recall that we are assuming (α, β) = (γ − δ, δ)), the term of lowest second coordinate on S(γ) ∩ |∆(g; x, y0, z0)| is

αps + pt(e − 1) pt (a, b) = , , ps ps

2 0 0 which is not in N since t < s, and is not (α, β) since e > 1. Thus (g; x, y , z ) is not (a, b) solvable, and

pt ept δ0 = < = β. ps ps

Remark 7.32. In Theorem 7.31 suppose that K has characteristic 0. From γ0 (7.18) we see that if {g}xy0z0 is solvable, then for 0 ≤ k ≤ r − 1 there exist i, j ∈ N with i = α(r − k), j = β(r − k) so that α, β ∈ N. Comparing (7.18) with the binomial expansion

r X r (z + ωxαyβ)r = (ωxαyβ)r−izi = zr + F, r − i i=0 we see that zr +F is solvable, a contradiction. Thus (γ, 0) is not solvable on |∆(g; x, y0, z)| if K has characteristic 0, and after well preparation, (γ, 0) is a vertex of |(g; x, y0, z0)|.

Example 7.33. With the notation of Theorem 7.31, it is possible to have (γ, 0) solvable in |∆(g; x, y0, z)| if K has positive characteristic, as is shown by the following simple example. Suppose that the characteristic p of K is greater than 5. Let g = zp + xyp2−1 + xp2−1y2.

The origin is isolated in Singp(g = 0), and (g; x, y, z) is very well pre- 1 1 pared. The vertices of |∆(g; x, y, z)| are (α, β) = (γ − δ, δ) = ( p , p − p ) and 128 7. Resolution of Surfaces in Positive Characteristic

1 2 0 p − p , p . Set y = y + ηx with 0 6= η ∈ K. Then g = zp + x(y0 + ηx)p2−1 + xp2−1(y0 + ηx)2 p Pp2−1 p2−1 p2−1−i p2−i 0 i = z + i=0 i η x (y ) +η2xp2+1 + 2ηy0xp2 + xp2−1(y0)2

0 1 1 The vertices of |∆(g; x, y , z)| are (α, β) = p , p − p and (γ, 0) = (p, 0). p− 1 However, (γ, 0) is solvable. Set z0 = z + η p xp. Then

 2  p −1 2 X p − 1 2 2 2 2 2 g = (z0)p+ ηp −1−ixp −i(y0)i +η2xp +1+2ηy0xp +xp −1(y0)2.  i  i=1 Since p2 − 1 ≡ −1 mod p, 1 1 0 0 it follows that δ1 = δxy0z0 (g) = p , and the vertices of |∆(g; x, y , z )| are 1 1 1 1 1 (α, β) = ( p , p − p ), (γ1 − δ1, δ1) = (p − p , p ) and (p + p , 0).

Theorem 7.34. For all n ∈ N, there are fn ∈ Rn such that fn is a generator of ISn,qn Rn, and good parameters (xn, yn, zn) for fn in Rn such that if

βn = βxnynzn (fn), n = xnynzn (fn),

αn = αxnynzn (fn), δn = δxnynzn (fn), 1 σ(n) = (βn, δn, , αn), n 1 1 then σ(n + 1) < σ(n) for all n in the lexicographical order in r! N × r! N × ( ∪ {∞}) × 1 . If β = β , δ = δ and 1 6= ∞, then Q+ r! N n+1 n n+1 n n 1 1 = − 1. n+1 n As an immediate corollary, we find a contradiction to the assumption that (7.8) has infinite length. We have now verified Theorem 7.9 for the final case τ(q) = 1.

Proof. We inductively construct regular parameters (xn, yn, zn) in Rn, a generator fn of ISn,qn Rn such that (xn, yn, zn) are good parameters for fn, Singr(fn) ⊂ V (xn, zn) or Singr(fn) ⊂ V (yn, zn) and (fn; xn, yn, zn) is well prepared. If Singr(fn) ⊂ V (xn, zn), we will further have that (fn; xn, yn, zn) is very well prepared.

Let f be a generator of IS,qR0. We ﬁrst choose (possibly formal) regular parameters (x, y, z) in R which are good for f = 0, such that |∆(f; x, y, z)| 7.3. τ(q) = 1 129 is very well prepared. By Lemma 7.16, z = 0 is an approximate manifold for f = 0. Let α = αx,y,z(f), β = βx,y,z(f).

By assumption, q is isolated in Singr(f). Suppose that fi and regular parameters (xi, yi, zi) in Ri have been de- ﬁned for i ≤ n as speciﬁed above.

If Singr(fn) = V (yn, zn), then Rn → Rn+1 must be a T4 transformation, 0 0 0 0 xn = xn+1, yn = yn+1, zn = yn+1zn+1. 0 0 0 Thus (fn+1; xn+1, yn+1, zn+1) is prepared by Lemma 7.29. Further, 0 0 Singr(fn+1) ⊂ V (yn+1, zn+1). 0 0 If Singr(fn+1) = V (xn+1, yn+1, zn+1), make a change of variables to very 0 0 well prepare. Otherwise, set yn+1 = yn+1, zn+1 = zn+1. We have βn+1 < βn by Lemma 7.29.

If Singr(fn) = V (xn, zn), then Rn → Rn+1 must be a T3 transformation

xn = xn+1, yn = yn+1, zn = xn+1zn+1, and (fn; xn, yn, zn) is very well prepared. Thus (fn+1; xn+1, yn+1, zn+1) is also very well prepared. Moreover, 1 1 (βn+1, δn+1, , αn+1) < (βn, δn, , αn) n+1 n by Lemma 7.27. Further, Singr(fn+1) ⊂ V (xn+1, zn+1).

Now suppose that Singr(fn) = V (xn, yn, zn). Then (f; xn, yn, zn) is very well prepared, and Rn+1 must be a T1 or T2 transformation of Rn.

If Rn+1 is a T2 transformation of Rn, 0 0 0 xn = xn+1yn+1, yn = yn+1, zn = yn+1zn+1, 0 0 then (fn+1; xn+1, yn+1, zn+1) is well prepared with βn+1 < βn by Lemma 0 0 7.29. Further, Singr(fn+1) ⊂ V (yn+1, zn+1). If 0 0 Singr(fn+1) = V (xn+1, yn+1, zn+1), then make a further change of variables to very well prepare. Otherwise, set 0 0 yn+1 = yn+1, zn+1 = zn+1. If Rn+1 is a T1 transformation of Rn, 0 0 xn = xn+1, yn = xn+1(yn+1 + η), zn = xn+1zn+1, 0 then Singr(fn+1) ⊂ V (xn+1, zn+1) by Lemma 7.5. If βn 6= 0 and η 6= 0 in the T1 transformation relating Rn and Rn+1, we can change variables to yn+1, zn+1 to very well prepare, with βn+1 < βn and Singr(fn+1) ⊂ V (xn+1, zn+1) by Theorem 7.31. 130 7. Resolution of Surfaces in Positive Characteristic

If η = 0 in the T1 transformation of Rn, and (xn+1, yn+1, zn+1) are 0 0 regular parameters of Rn+1 obtained from (xn+1, yn+1, zn+1) by very well preparation, then (fn+1; xn+1, yn+1, zn+1) is very well prepared, 1 1 (βn+1, δn+1, , αn+1) < (βn, δn, , αn), n+1 n and Singr(fn+1) ⊂ V (xn+1, zn+1) by Lemma 7.28. 0 If βn = 0, we can make a translation yn = yn + ηxn, and have that 0 0 (gnxn, yn, zn) is very well prepared, and βn, γn, δn, n, αn are unchanged. Thus we are in the case of η = 0.

7.4. Remarks and further discussion

The first proof of resolution of singularities of surfaces in positive characteristic was given by Abhyankar ([1], [3]). The proof used valuation-theoretic methods to prove local uniformization (defined in Section 2.5). The most general proof of resolution of surfaces, which includes excellent surfaces, was given by Lipman [63]. This lucid article gives a self-contained geometric proof. This proof of resolution of surfaces which are hypersurfaces over an algebraically closed field (Theorem 7.1) extends to give a proof of resolution of excellent surfaces. Some general remarks about the proof are given in the final 3 pages of Hironaka’s notes [53]. The first part of our proof of Theorem 7.1 generalizes to the case of an excellent surface S, and we reduce to proving a generalization of Theorem 7.9, when V = spec(R), where R is a regular local ring with maximal ideal m and S = spec(R/I) is a two-dimensional equidimensional local ring with Singr(R/I) = V (m). There are two main new points which must be considered. The first point is to find a generalization of the polygon |∆(f; x, y, z)| to the case ∼ where OˆS,q = R/I is as above. This is accomplished by the general theory of characteristic polyhedra [54]. The second point is to extend the theory to the case of a non-closed residue field. This is accomplished in [23]. We will now restrict to the case when our surface S is a hypersurface over an arbitrary field K. The proof of Theorem 7.9 in the case when τ(q) = 3 is the same, and if τ(q) = 2, the proof is really the same, since we must then have equality of residue fields K(qi) = K(qi+1) for all i in the sequence

(7.3), under the assumption that τ(qi) = 2, νqi (Si) = r for all i. This is the analogue of our proof of resolution of curves embedded in a non-singular surface, over an arbitrary ﬁeld. In the case when τ(q) = 1, some extra care is required. It is possible to have non-trivial extensions of residue ﬁelds k(qi) → k(qi+1) in the sequence 7.4. Remarks and further discussion 131

(7.8). However, if k(qi+1) 6= k(qi), we must have βi+1 < βi (This is proven in the theorem on page 26 of [23]). Thus (β , δ , 1 , α ) drops after every transformation in (7.8), and we see i i i i that we must eventually have ν drop or τ increase. The essential difficulty in resolution in positive characteristic, as opposed to characteristic zero, is the lack of existence of a hypersurface of maximal contact (see Exercise 7.35 at the end of this section). Abhyankar is able to finesse this difficulty to prove resolution for 3- dimensional varieties X, over an algebraically closed field K of characteristic p > 5 (13.1, [4]). We reduce to proving local uniformization by a patching argument (9.1.7, [4]). The starting point of the proof of local uniformization is to use a projection argument (12.4.3, 12.4.4 [4]) to reduce the problem to the case of a point on a normal variety of multiplicity ≤ 3! = 6. A simple exposition of this projection method is given on page 200 of [62]. We will sketch the proof of resolution, in the case of a hypersurface singularity of multiplicity ≤ 6. We then have that a local equation of this singularity is r r−1 (7.19) w + ar−1(x, y, z)w + ··· + a0(x, y, z) = 0. 1 Since r ≤ 6 < char(K), it follows that r ∈ K, and we can perform a Tschirnhausen transformation to reduce to the case r r−2 w + ar−2(x, y, z)w + ··· + a0(x, y, z) = 0. Now w = 0 is a hypersurface of maximal contact, and we have a good theorem for embedded resolution of 2-dimensional hypersurfaces, so we reduce to the case where each ai is a monomial in x, y, z times a unit. Now we have reduced to a combinatorial problem, which can be solved in a characteristic free way. A resolution in the case where (7.19) has degree p = char K is found by Cossart [24]. The proof is extraordinarily difficult. Suppose that X is a variety over a field K. In [36], de Jong has shown that there is a dominant proper morphism of K-varieties X0 → X such that dim X0 = dim X and X0 is non-singular. If K is perfect, then the finite extension of function fields K(X) → K(X0) is separable. This is weaker than a resolution of singularities, since the map is in general not birational (the extension of function fields K(X0) → K(X) is finite). This proof relies on sophisticated methods in the theory of moduli spaces of curves. Exercise 7.35. 1. (Narasimhan [69]) Let K be an algebraically closed field of char- 4 acterisitc 2, and let X be the 3-fold in AK with equation f = w2 + xy3 + yz3 + zx7 = 0. 132 7. Resolution of Surfaces in Positive Characteristic

a. Show that the maximal multiplicity of a singular point on X is 2, and the locus of singular points on X is the monomial curve C with local equations y3 + zx6 = 0, xy2 + z3 = 0, yz2 + x7 = 0, w2 + zx7 = 0, which has the parameterization t → (t7, t19, t15, t32). Thus C has embedding dimension 4 at the origin, so there cannot exist a hypersurface (or formal hypersurface) of maximal contact for X at the origin. b. Resolve the singularities of X. 2. (Hauser [50]) Let K be an algebraically closed field of characteristic 2 2 4 2, and let S be the surface in AK with equation f = x + y z + y2z4 + z7 = 0. a. Show that the maximal multiplicity of points on S is 2, and the singular locus of S is defined by the singular curve y2 +z3 = 0, x + yz2 = 0. b. Suppose that X is a hypersurface on a non-singular variety W , and p ∈ X is a point in the locus of maximal multiplicity r of X. A hypersurface H through p is said to have permanent contact with X if under any sequence of blow-ups π : W1 → W of W , with non-singular centers, contained in the locus of points of multiplicity r on the strict transform of X, the strict transform of H contains all points of the intersection of the strict transform of X with multiplicity r which are in the fiber π−1(p). Show that there does not exist a non-singular hypersurface of permanent contact at the origin for the above surface S. Conclude that a hypersurface of maximal contact does not exist for S. c. Resolve the singularities of S. Chapter 8

Local Uniformization and Resolution of Surfaces

In this chapter we present a proof of resolution of surface singularities through local uniformization of valuations. Our presentation is a modernization of Zariski’s original proof in [86] and [88]. An introduction to this approach was given earlier in Section 2.5

8.1. Classiﬁcation of valuations in function ﬁelds of dimension 2

Let L be an algebraic function field of dimension two over an algebraically closed field K of characteristic zero. That is, L is a finite extension of a rational function field in two variables over K. A valuation of the function field L is a valuation ν of L which is trivial on K. ν is a homomorphism ν : L∗ → Γ from the multiplicative group of L onto an ordered abelian group Γ such that 1. ν(ab) = ν(a) + ν(b) for a, b ∈ L∗, 2. ν(a + b) ≥ min{ν(a), ν(b)} for a, b ∈ L∗, 3. ν(c) = 0 for 0 6= c ∈ K. We formally extend ν to L by setting ν(0) = ∞. We will assume throughout this section that ν is non-trivial, that is, Γ 6= 0. Some basic references to the valuation theory of algebraic function fields are [92] and [3].

133 134 8. Local Uniformization and Resolution of Surfaces

A basic invariant of a valuation is its rank. A subgroup Γ0 of Γ is said to be isolated if it has the property that if 0 ≤ α ∈ Γ0 and β ∈ Γ is such that 0 ≤ β < α, then β ∈ Γ0. The isolated subroups of Γ form a single chain of subgroups. The length of this chain is the rank of the valuation. Since L is a two-dimensional algebraic function field, we have that rank(ν) ≤ 2 (cf. the corollary and the note to the lemma of Section 10, Chapter VI of [92]). If Γ is of rank 1 then Γ is embeddable as a subgroup of R (cf. page 45 of Section 10, Chapter VI [92]). If Γ is of rank 2, then there is a non-trivial isolated subgroup Γ1 ⊂ Γ such that Γ1 and Γ/Γ1 are embeddable as subgroups of R. If these subgroups of R are discrete, then ν is called a discrete valuation. In particular, a discrete valuation (in an algebraic function field of dimension two) can be of rank 1 or of rank 2. The ordered chain of prime ideals p of V corresponds to the isolated subgroups Γ0 of Γ by P = {f ∈ V | ν(f) ∈ Γ − Γ0} ∪ {0} (cf. Theorem 15, Section 10, Chapter VI [92]). In our analysis, we will find a rational function field of two variables L0 over which L is finite, and consider the restricted valuation ν0 = ν | L0. We will consider L0 = K(x, y), where x, y ∈ L are algebraically independent elements such that ν(x) and ν(y) are non-negative. Let [L0 : L] = r. Suppose that w ∈ L, and suppose that r (8.1) F (x, y, w) = A0(x, y)w + ··· + Ar(x, y) = 0 is the irreducible relation satisfied by w, with Ai ∈ K[x, y] for all i. Since r−i r−j ν(F ) = ∞, there must be two distinct terms Aiw , Ajw of the same value, where we can assume that i < j. Thus (j − i)ν(w) = ν( Aj ), and we Ai have that r!Γ0 ⊂ Γ. Thus Γ and Γ0 have the same rank, and Γ is discrete if and only if Γ0 is discrete. Associated to the valuation ν is the valuation ring V = {f ∈ L | ν(f) ≥ 0}. V has a unique maximal ideal

mV = {f ∈ V | ν(f) > 0}.

The residue field K(V ) = V/mV of V contains K and has transcendence degree r∗ ≤ r over K. ν is called an r∗-dimensional valuation. In our case (ν non-trivial) we have r∗ = 0 or 1. If R is local ring contained in L, we will say that ν dominates R if R ⊂ V and mV ∩ R is the maximal ideal of R. If S is a subring of V , we will say that the center of ν on S is the prime ideal mV ∩ S. We will say that R is an algebraic local ring of L if R is essentially of finite type over K (R is a localization of a finite type K-algebra) and R has quotient field L. 8.1. Classification of valuations in function fields of dimension 2 135

We again consider our algebraic extension L0 = K(x, y) → L. The 0 0 valuation ring of ν is V = V ∩ L, with maximal ideal mV 0 = mV ∩ L. The residue ﬁeld K(V 0) is thus a subﬁeld of K(V ). Suppose that w∗ ∈ K(V ), and let w be a lift of w∗ to V . We have an irreducible equation of algebraic dependence (8.1) of w over K[x, y]. Let Ah be such that

ν(Ah) = min{ν(A1), . . . , ν(Ar)}.

We may assume that h 6= r, because if ν(Ar) < ν(Ai) for i < r, then we r−i would have ν(Ar) < ν(Aiw ) for 1 ≤ i ≤ r − 1, since ν(w) ≥ 0, so that Ai ν(F ) = ν(Ar) < ∞, which is impossible. Thus ν( ) ≥ 0 for 0 ≤ i ≤ r, so Ah that Ai ∈ V 0 for all i. Let a∗ be the corresponding residues in K(V 0). We Ah i have a relation ∗ ∗ r ∗ ∗ r−h+1 ∗ r−h ∗ ∗ r−h−1 ∗ a0(w ) + ··· + ah−1(w ) + (w ) + ah+1(w ) + ··· + ar = 0. Since r − h > 0, w∗ is algebraic over K(V 0). Thus ν and ν0 = ν | L0 have the same dimension. We will now give a classiﬁcation of the various types of valuations which occur on L.

8.1.1. One-dimensional valuations. Let x ∈ L be the lift of an element of K(V ) which is transcendental over K. Suppose that every y ∈ L which is transcendental over K(x) satisfies ν(y) = 0. Then ν must be trivial when restricted to K(x, y) for any y which is transcendental over K(x). Since L is finite over K(x, y), we would have that ν has rank 0 on L, so that ν is trivial on L, a contradiction. We may thus choose y ∈ L such that x, y are algebraically independent over K, ν(x) = 0, ν(y) > 0 and the residue of x in K(V ) is transcendental over K. Let L0 = K(x, y), and let ν0 = ν | L0 0 0 with valuation ring V = V ∩ L . By construction, if mV 0 is the maximal 0 ideal of V , there exists c ∈ K such that mV 0 ∩ K[x, y] ⊂ (x − c, y). Suppose (if possible) that p = mV 0 ∩ K[x, y] = (x − c, y). Then the residue of x in K(V ) is c, a contradiction. Thus p is a height one prime in K[x, y] and 0 p = (y). It follows that the valuation ring V is Rp (cf. Theorem 9, Section 0 5, Chapter VI [92]) and the value group of ν is Z. The valuation ring V is a localization of the integral closure of the local Dedekind domain V 0 in L. Thus V is itself a local Dedekind domain which is a localization of a ring of finite type over K. In particular, V is itself a regular algebraic local ring of L.

8.1.2. Zero-dimensional valuations of rank 2. Let Γ1 be the non- trivial isolated subgroup of Γ, with corresponding prime ideal p ⊂ V . The composition of ν with the order-preserving homomorphism Γ → Γ/Γ1 de- ﬁnes a valuation ν1 of L of rank 1. Let V1 be the valuation ring of ν1. V1 consists of the elements of K whose value by ν is in Γ1 or is positive. Hence 136 8. Local Uniformization and Resolution of Surfaces

V1 = Vp. ν induces a valuation ν2 of K(Vp) with value group Γ1, where ∗ ∗ ν2(f) = ν(f ) if f ∈ Vp is a lift of f to Vp. Since K(Vp) has a non-trivial valuation, which vanishes on K, K(Vp) cannot be an algebraic extension of K. Thus K(Vp) has positive transcendence degree over K, so it must have ∗ ∗ transcendence degree 1. Let x ∈ K(Vp) be such that x is transcendental ∗ over K. Let x be a lift of x to Vp. Then K(x) ⊂ Vp, so that ν1 is trivial on K(x), and ν1 is a valuation of the one-dimensional algebraic function ﬁeld L over K(x).

Let V2 ⊂ K(VP ) be the valuation ring of ν2, and let π : Vp → K(Vp) −1 be the residue map. Then V = π (V2). ν1 is a 1-dimensional valuation of L, so Vp is an algebraic local ring which is a local Dedekind domain. The residue ﬁeld K(Vp) is an algebraic function ﬁeld of dimension 1 over K, so V2 is also a local Dedekind domain. Thus Γ is a discrete group. Let w be a regular parameter in Vp.

Let w be a regular parameter in Vp. Given any non-zero element f ∈ L, m we have a unique expression f = w g, where m = ν1(f) ∈ Z, and g is a ∗ ∗ unit in Vp. Let g be the residue of g in K(Vp). Let n = ν2(g ) ∈ Z. Then, 2 up to an order-preserving isomorphism of Γ, ν(f) = (m, n) ∈ Γ = Z , where 2 Z has the lexicographic order.

8.1.3. Discrete zero-dimensional valuations of rank 1. Let x ∈ L be such that ν(x) = 1. If y ∈ L and ν(y) = n1, then there exists a unique n c1 ∈ K such that n2 = ν(y − c1x 1 ) > ν(y). If we iterate this construction inﬁnitely many times, we have a uniquely determined Laurent series expansion

n1 n2 c1x + c2x + ...

n nm with ν(y − c1x 1 − · · · − cmx ) = nm+1 > nm for all m. This construction deﬁnes an embedding of K-algebras L ⊂ Khhxii, where Khhxii is the ﬁeld of Laurent series in x. The order valuation on Khhxii restricts to the valuation ν on L.

8.1.4. Non-discrete zero-dimensional valuations of rank 1. We sub- divide this case by the rational rank of ν. The rational rank is the dimension of Γ ⊗ Q as a rational vector space. Since ν has rank 1, we may assume that Γ ⊂ R. If 1 6∈ Γ, we can choose 0 < λ ∈ Γ and deﬁne an order-preserving 1 α group isomorphism φ :Γ → λ Γ by φ(α) = λ . Then ν = φ ◦ ν is a valuation of L with the same valuation ring {f ∈ L | ν(f) > 0} = {f ∈ L | ν(f) > 0} = V as ν. We may thus normalize ν (by replacing ν with the equivalent valuation ν) so that Γ is an ordered subgroup of R and 1 ∈ Γ. 8.2. Local uniformization of algebraic function ﬁelds of surfaces 137

Suppose that ν has rational rank larger that 1. Choose x, y ∈ L such that ν(x) = 1 and ν(y) = τ is a positive irrational number. Suppose that P i j F (x, y) = aijx y is a polynomial in x and y with coefficients in K. Then i j ν(aijx y ) = i + jτ for any term with aij 6= 0, so all the monomials in the expansion of F have distinct values. It follows that ν(F ) = min{i + τj | aij 6= 0}. We conclude that x and y are algebraically independent, since a relation F (x, y) = 0 has infinite value. Furthermore, if L1 = K(x, y) and ν1 = ν | L1, we have that the value group Γ1 of ν1 is Z + Zτ. Since Γ/Γ1 is a finite group, we have that Γ is a group of the same type. In particular, Γ has rational rank 2. Now suppose that ν has rational rank 1. We can normalize Γ so that it is an ordered subgroup of Q, whose denominators are not bounded, as Γ is not discrete. In Example 3, Section 15, Chapter VI [92], examples are given of two-dimensional algebraic function fields with value group equal to any given subgroup of the rationals. There is a nice interpretation of this kind of valuation in terms of certain “formal” Puiseux series (cf. MacLane and Shilling [64]) where the denominators of the rational exponents in the Puiseux series expansion are not bounded. Exercise 8.1. Let K be a field, R = K[x, y] a polynomial ring and L the quotient field of R. 1. Show that the rule ν(x) = 1, ν(y) = π defines a K-valuation on L which satisfies

ν(f) = min{i + jπ | aij 6= 0} P i j if f = aijx y ∈ K[x, y]. 2. Compute the rank, rational rank and value group Γ of ν. 3. Let V be the valuation ring of ν. Suppose that τ ∈ Γ. Show that

Iτ = {f ∈ V | ν(f) ≥ τ} is an ideal of V . 4. Show that V is not a Noetherian ring.

8.2. Local uniformization of algebraic function ﬁelds of surfaces

In this section, we prove the following local uniformization theorem. Theorem 8.2. Suppose that K is an algebraically closed field of characteristic zero, L is a two-dimensional algebraic function field over K and ν is a valuation of L. Then there exists a regular local ring A which is essentially of finite type over K with quotient field L such that ν dominates A. 138 8. Local Uniformization and Resolution of Surfaces

Our proof is by a case by case analysis of the different types of valuations of L which were classified in the previous section. If ν is 1-dimensional, we have that R = V satisfies the conclusions of the theorem. We must prove the theorem in the case when ν has dimension zero, and one of the following three cases hold: ν has rational rank 1, or ν has rank 1 and rational rank 1, or ν has rank 2. The case when ν has rank 1 includes the cases when ν is discrete and ν is not discrete. We will follow the notation of the preceeding section.

Suppose that K[x1, . . . , xn] is a polynomial ring, and m = (x1, . . . , xn). For f ∈ K[x1, . . . , xn], ord(f) will denote the order of f in K[x1, . . . , xn]m (cf. Section A.4). In particular, ord(f) = r if and only if f ∈ mr and f 6∈ mr+1.

8.2.1. Valuations of rational rank 1 and dimension 0.

Lemma 8.3. Suppose that S = K[x1, . . . , xn] is a polynomial ring, ν is a rank 1 valuation with valuation ring V and maximal ideal mV which contains S such that K(V ) = K and mV ∩ S = (x1, . . . , xn). Suppose that there exist fi ∈ S for i ∈ N such that

ν(f1) < ν(f2) < ··· < ν(fn) < ··· is a strictly increasing sequence. Then limi→∞ ν(fi) = ∞.

Proof. Suppose that ρ > 0 is a positive real number. We will show that there are only finitely many real numbers less than ρ which are the values of elements of S. Let τ = min{ν(f) | f ∈ (x1, . . . , xn)}. τ is well defined since S is Noetherian. In fact, if there were not a minimum, we would have an infinite descending sequence of positive real numbers

τ1 > τ2 > ··· τi > ··· and elements gi ∈ (x1, . . . , xn) such that ν(gi) = τi. Let

Iτi = {f ∈ S | ν(f) ≥ τi}.

The Iτi are distinct ideals of S which form a strictly ascending chain

Iτ1 ⊂ Iτ2 ⊂ · · · ⊂ Iτi ⊂ · · · , which is impossible. ρ Choose a positive integer r such that r > τ . If f ∈ S is such that r ν(f) < ρ, then write f = g + h with h ∈ (x1, . . . , xn) and deg(g) < r. ν(h) > ρ implies ν(g) = ν(f). Thus there exist a ﬁnite number of elements g1, . . . , gm ∈ S (the monomials of degree < r) such that if 0 < λ < ρ is the value of an element of S, then there exist c1, . . . , cm ∈ K such that

(8.2) ν(c1g1 + ··· + cmgm) = λ. 8.2. Local uniformization of algebraic function ﬁelds of surfaces 139

We will replace the gi with appropriate linear combinations of the gi so that

ν(g1) < ν(g2) < ··· < ν(gm).

Then λ = ν(gk) in (8.2), where ck is the ﬁrst non-zero coeﬃcient.

By induction, it suﬃces to make a linear change in the gi so that

(8.3) ν(g1) < ν(gi) if i > 1.

After reindexing the gi, we can suppose that there exists an integer l > 1 such that

ν(g1) = ν(g2) = ··· = ν(gl) and

ν(gi) > ν(g1) if l < i. The equality ν( gi ) = 0 for 2 ≤ i ≤ l implies gi ∈ V and there exist g1 g1 c ∈ K(V ) = K such that gi has residue c in K(V ). Then gi − c ∈ m i g1 i g1 i V implies ν( gi − c ) > 0, so that ν(g − c g ) > ν(g ) for 2 ≤ i ≤ l. After g1 i i i 1 1 replacing gi with gi − cig1 for 2 ≤ i ≤ l, we have that (8.3) is satisfied. Theorem 8.4. Suppose that S = K[x, y] is a polynomial ring over an algebraically closed field K of characteristic zero, ν is a rational rank 1 valuation of K(x, y) with valuation ring V and maximal ideal mV , K(V ) = V/mV = K such that S ⊂ V and the center of V on S is the maximal ideal (x, y). Further suppose that f ∈ K[x, y] is given. Then there exists a birational extension K[x, y] → K[x0, y0] (K[x, y] and K[x0, y0] have a common quotient field) such that K[x0, y0] ⊂ V , the center of ν on K[x0, y0] is (x0, y0), and f = (x0)lδ, where l is a non-negative integer and δ ∈ K[x0, y0] is not in (x0, y0).

Proof. Set r = ord f(0, y). We have 0 ≤ r ≤ ∞. If r = 0 we are done, so suppose that r > 0. We have an expansion d X αi βi X αi βi (8.4) f = x γi(x)y + x γi(x)y , i=1 i>d α β where the ﬁrst sum is over terms with minimal value ρ = ν(x i γi(x)y i ) for 1 ≤ i ≤ d, γi(x) ∈ K[x] are polynomials with non-zero constant term, α β β1 < ··· < βd, βd+1 < βd+2 < ··· and ν(x i γi(x)y i ) > ρ if i > d. ν(x) a Set ν(y) = b with a, b ∈ N,(a, b) = 1. There exist non-negative integers 0 0 0 0 ya a , b such that ab − ba = 1. Then ν( xb ) = 0 implies there exists 0 6= c ∈ ya K(V ) = K such that ν( xb − c) > 0. Set a a0 b b0 x = x1(y1 + c) , y = x1(y1 + c) . 140 8. Local Uniformization and Resolution of Surfaces

We have that b0 −a0 −b a x1 = x y , y1 + c = x y , so that ν(y1) > 0 and ν(y) ν(y) ν(x ) = [ab0 − a0b] = > 0. 1 b b Set α = α1a + β1b. We have that αia + βib = α for 1 ≤ i ≤ d. Thus there exist ei ∈ Z such that a b αi − α1 0 0 0 = a b βi − β1 ei for 1 ≤ i ≤ d. By Cramer’s rule, we have a 0 βi − β1 = Det 0 = aei. a ei

βi−β1 Thus ei = a . α We have a factorization f = x1 f1(x1, y1), where d 0 0 βi−β1 a α1+b β1 X f1(x1, y1) = (y1 + c) ( γi(y1 + c) a ) + x1Ω i=1 is the strict transform of f in the birational extension K[x1, y1] of K[x, y]. We have r1 < ∞. ord f(0, y) = r implies there exists an i such that αi = 0, r βi = r. Thus ν(y ) ≥ ρ, and βi ≤ r for 1 ≤ i ≤ d. We have β − β r = ord f (0, y ) ≤ d 1 ≤ r. 1 1 1 a If r1 < r, then we have a reduction, and the theorem follows from induction on r, since the conclusions of the theorem hold in K[x, y] if r = 0. We thus assume that r1 = r. Then βd = r, β1 = 0, a = 1 and αd = 0. Further, there is an expression d X βi r γi(0)(y1 + c) = γd(0)y1. i=1 Let u be an indeterminate, and set d X βi r g(u) = γi(0)u = γd(0)(u − c) . i=1

The binomial theorem implies that βd−1 = r − 1, and thus ν(y) = ν(xαd−1 ).

y αd−1 Let 0 6= λ ∈ k(V ) = k be the residue of xαd−1 . Then ν(y − λx ) > ν(y). Set y0 = y − λxαd−1 and f 0(x, y0) = f(x, y). We have ord f 0(0, y0) = ord f(0, y) = r. 8.2. Local uniformization of algebraic function ﬁelds of surfaces 141

yr is a minimal value term of f(x, y) in the expansion (8.4), so ν(y) ≤ ν(f). Replacing y with y0 and f with f 0 and iterating the above procedure after (8.4), either we achieve a reduction ord f1(0, y1) < r, or we ﬁnd that there exists a0(x) ∈ K[x] such that ν(y0 − a0(x)) > ν(y0). We further have that ν(y0) ≤ ν(f). Set y00 = y0 − a0(x) and repeat the algorithm following (8.4) until we either reach a reduction ord f1(0, y1) < r or show that there (i) exist polynomials a (x) ∈ K[x] for i ∈ N such that we have an inﬁnite increasing bounded sequence ν(a(x)) < ν(a0(x)) < ··· . Since each ν(a(i)(x)) is an integral multiple of ν(x), this is impossible. Thus there is a polynomial q(x) ∈ K[x] such that after replacing y with y − q(x), and following the algorithm after (8.4), we achieve a reduction r1 < r in K[x1, y1]. By induction on r, we can achieve the conclusions of the theorem.

We now give the proof of Theorem 8.2 in the case of this subsection, that is, ν has dimension 0 and rational rank 1. Let x, y ∈ L be algebraically independent over K and of positive value. Let w be a primitive element of L over K(x, y). We can assume that w has positive value, for if ν(w) = 0, there exists c ∈ K such that the residue of w − c is non-zero in K(V ) = K, and we can replace w with w − c. Let R = K[x, y, w]. Then R ⊂ V , the center of ν on R is the maximal ideal (x, y, w), and R has quotient ﬁeld L. Let z be algebraically independent over K(x, y), and let f(x, y, z) in the polynomial ring K[x, y, z] be the irreducible polynomial such that R =∼ K[x, y, z]/(f) (with w mapping to z + (f)). After possibly making a change of variables, replacing x with x + βz and y with y + γz for suitable β, γ ∈ K, we may assume that r = ord f(0, 0, z) < ∞.

If r = 1, then RmV ∩R is a regular local ring which is dominated by ν (by Exercise 3.4), so the conclusions of the theorem are satisﬁed. We thus suppose that r > 1. Let π : K[x, y, z] → K[x, y, w] be the natural surjection. For g ∈ K[x, y, z], we will denote ν(π(g)) by ν(g). Write

d X σi X σi (8.5) f(x, y, z) = ai(x, y)z + ai(x, y)z , i=1 i>d σ where ai(x, y) ∈ K[x, y] for all i, the ai(x, y)z i are the terms of minimal σ σ value ρ = ν(ai(x, y)z i ) for 1 ≤ i ≤ d, ν(ai(x, y)z i ) > ρ for i > d and σ1 < σ2 < ··· < σd. 142 8. Local Uniformization and Resolution of Surfaces

By Theorem 8.4, there exists a birational extension K[x, y] → K[x1, y1] with K[x1, y1] ⊂ V and mV ∩ K[x1, y1] = (x1, y1) such that d X λi σi X λi σi f = x1 ai(x1, y1)z + x1 ai(x1, y1)z , i=1 i>d where ai(0, 0) 6= 0 for all i. R1 = K[x1, y1, z1]/(f) is a domain with quotient ﬁeld L, R → R1 is birational, R1 ⊂ V and mV ∩ R1 = (x1, y1, z). Write ν(z) = s with s, t relatively prime positive integers. Let s0, t0 be positive ν(x1) t integers such that s0t − st0 = 1. Set t t0 s s0 x1 = x2(z2 + c2) , y1 = y2, z = x2(z2 + c2) , where 0 6= c2 ∈ K is determined by the condition ν(z2) > 0. Let α = λ1t + σ1s. We have α = λit + σis for 1 ≤ i ≤ d. We have (as in the proof σi−σ1 of Theorem 8.4) that t is a non-negative integer for 1 ≤ i ≤ d and α f = x2 f2(x2, y2, z2), where d 0 0 σi−σ1 t λ1+s σ1 X f2(x2, y2, z2) = (x2 + c2) ( ai(z2 + c2) t ) + x2Ω i=1 is the strict transform of f in k[x2, y2, z2]. Set R2 = K[x2, y2, z2]/(f2). Then R1 → R2 is birational, R2 ⊂ V and mV ∩ R2 = (x2, y2, z2). Set σ − σ r = ord f (0, 0, z ) ≤ d 1 ≤ r. 1 2 2 t If r1 < r, we have a reduction, so we may assume that r1 = r. Then we have σd = r, σ1 = 0, t = 1 and ad(0, 0) 6= 0. We further have a relation d X σi r ai(0, 0)(z2 + c2) = ad(0, 0)z2. i=1 Set d X σi r g(u) = ai(0, 0)u = ad(0, 0)(u − c2) . i=1 The binomial theorem shows that σd−1 = r − 1 and ν(z) = ν(ad−1(x, y)). r−1 There exists c ∈ k such that ν(z − cad−1(x, y)) > ν(z). Then radz is a ∂f ∂f 0 minimal value term of ∂z , so that ν( ∂z ) ≥ ν(z). Set z = z − cad−1(x, y), 0 0 0 0 ∂f 0 ∂f f (x, y, z ) = f(x, y, z). We have ord(f (0, 0, z )) = r and ∂z0 = ∂z . We repeat the analysis following (8.5), with f replaced by f 0 and z 0 replaced with z . Either we get a reduction r1 = ord(f2(0, 0, z2)) < r, or there exists a0(x, y) ∈ K[x, y] such that ν(z0 − a0(x, y)) > ν(z0) and ∂f 0 ∂f ν(z0) ≤ ν( ) = ν( ). ∂z0 ∂z 8.2. Local uniformization of algebraic function ﬁelds of surfaces 143

Set z00 = z0 −a0(x, y) and repeat the algorithm following (8.5). If we do reach a reduction r1 < r after a ﬁnite number of iterations, there exist polynomials (i) a (x, y) ∈ K[x, y] for i ∈ N such that ν(a(x, y)) < ν(a0(x, y)) < ··· is an increasing bounded sequence of real numbers. By Lemma 8.3 this is impossible. Thus there exists a polynomial a(x, y) ∈ K[x, y] such that after replacing z with z −a(x, y), the algorithm following (8.5) results in a reduction r1 < r. We then repeat the algorithm, so that by induction on r, we construct a birational extension of local rings R → R1 so that R1 is a regular local ring which is dominated by V .

8.2.2. Valuations of rank 1, rational rank 2 and dimension 0. After normalizing ν, we have that its value group is Z + Zτ, where τ is a positive irrational number. Choose x, y ∈ L such that ν(x) = 1 and ν(y) = τ. By the analysis of Subsection 8.1.4, x and y are algebraically independent over K. Let w be a primitive element of L over K(x, y) which has positive value. There exists an irreducible polynomial f(x, y, z) in the polynomial ring K[x, y, z] such that T = K[x, y, w] =∼ K[x, y, z]/(f) (where w maps to m z +(f)) and mV ∩T = (x, y, w). After possibly replacing x with x+βz and y with y +γzm, where m is suﬃciently large that mν(w) > max{ν(x), ν(y)}, and β, γ ∈ K are suitably chosen, we may assume that r = ord(f(0, 0, z)) < ∞.

If r = 1, TmV ∩T is a regular local ring (by Exercise 3.4), so we assume that r > 1. Let π : K[x, y, z] → K[x, y, w] be the natural surjection. For g ∈ K[x, y, z], we will denote ν(π(g)) by ν(g). Write d X αi βi γi X αi βi γi f = aix y z + aix y z , i=1 i>d α β γ where all ai ∈ K are non-zero, aix i y i z i for 1 ≤ i ≤ d are the minimal value terms, and γ1 < γ2 < ··· < γd. There exist integers s, t such that ν(z) = s + τt, and there exist integers M,N such that ν(xαi yβi zγi ) = M + Nτ for 1 ≤ i ≤ d. We thus have equalities M = α + sγ = ··· = α + sγ , (8.6) 1 1 d d N = β1 + tγ1 = ··· = βd + tγd. We further have

(αi + sγi) + (βi + tγi)τ > M + Nτ 144 8. Local Uniformization and Resolution of Surfaces

α β γ if i > d. Since ord(f(0, 0, z)) = r and adx d y d z d is a minimal value term, we have γd ≤ r. We expand τ into a continued fraction: 1 τ = h1 + . h + 1 2 h3+··· Let fj be the convergent fractions of τ (cf. Section 10.2 [46]). Since lim fj = gj gj τ, we have fj fj (αi + sγi) + (βi + tγi) > M + N gj gj for j = p − 1 and j = p, whenever i > d and p is suﬃciently large. We further have that sgp + tfp > 0 and sgp−1 + tfp−1 > 0 for p suﬃciently large. s t z Since ν(x y ) = ν(z), there exists c ∈ K = K(V ) such that ν( xsyt − c) > 0. Consider the extension

(8.7) K[x, y, z] → K[x1, y1, z1], where gp gp−1 x = x1 y1 , fp fp−1 (8.8) y = x1 y1 , s t sgp+tfp sgp−1+tfp−1 z = x y (z1 + c) = x1 y1 (z1 + c). We have f g − f g = = ±1 (cf. Theorem 150 [46]), and , −τ + fp−1 p−1 p p p−1 gp−1 and τ − fp have the same signs (cf. Theorem 154 [46]). From (8.8), we see gp that fp−1 gp x y x1 = , y1 = , ygp−1 xfp from which we deduce that (8.7) is birational, and

fp−1 fp ν(x1) = gp−1( − τ), ν(y1) = gp(τ − ). gp−1 gp

Hence ν(x1), ν(y1), ν(z1) are all > 0. We have

Mgp+Nfp Mgp−1+Nfp−1 f(x, y, z) = x1 y1 f1(x1, y1, z1), where d γ1 X γi−γ1 f1 = (z1 + c) ( ai(z1 + c) ) + x1y1H i=1 is the strict transform of f in K[x1, y1, z1]. We thus have constructed a birational extension T → T1 = K[x1, y1, z1]/(f1) such that T1 ⊂ V and mV ∩ T1 = (x1, y1, z1).

We have f1(0, 0, 0) = 0, so u = c is a root of d X γi−γ1 φ(u) = aiu = 0. i=1 8.2. Local uniformization of algebraic function ﬁelds of surfaces 145

Let r1 = ord f1(0, 0, z1). Then r1 ≤ γd − γ1 ≤ r. If r1 < r, we have obtained a reduction, so we may assume that r1 = r. Thus γ1 = 0, γd = r, αd = βd = 0 (since γi ≤ r for 1 ≤ i ≤ d) and r φ(u) = ad(u − c) .

Hence γd−1 = r − 1 and ad−1 6= 0, by the binomial theorem. We have ν(z) = αd−1+βd−1τ. Our conclusion is that there exist non-negative integers m and n such that ν(z) = m + nτ. There exists c ∈ K such that ν(z − cxmyn) > ν(z). We make a change of variables in K[x, y, z], replacing z with z0 = z −cxmyn, and let f 0(x, y, z0) = f(x, y, z). We have ν(z0) > ν(z) and ord f 0(0, 0, z0) = r. r−1 ∂f ∂f Since radz is a minimal value term of ∂z , we have ν(z) ≤ ν( ∂z ). We ∂f ∂f 0 further have ∂z = ∂z0 . We now apply a transformation of the kind (8.8) with 0 respect to the new variables x, y, z . If we do not achieve a reduction r1 < r, 0 we have a relation ν(z ) = m1 + n1τ with m1, n1 non-negative integers, and we have ∂f ν(z) < ν(z0) ≤ ν( ). ∂z We have that there exists c2 ∈ K such that

0 m1 n1 0 ν(z − c2x y ) > ν(z ). 00 0 m n Set z = z − c2x 1 y 1 . We now iterate the procedure starting with the transformation (8.8). If we do not achieve a reduction r1 < r after a ﬁnite ∂f number of iterations, we construct an inﬁnite bounded sequence (since ∂z 6= 0 in T )

m + nτ < m1 + n1τ < ··· < mi + niτ < ··· , where mi, ni are non-negative integers, which is impossible. We thus must achieve a reduction r1 < r after a ﬁnite number of interations. By induction on r, we can construct a birational extension T → T1 such that T1 ⊂ V and

(T1)mV ∩T1 is a regular local ring.

8.2.3. Valuations of rank 2 and dimension 0. Suppose that ν has rank 2 and dimension 0. Let 0 ⊂ p ⊂ mV be the distinct prime ideals of V . Let x, y ∈ L be such that x, y are a transcendence basis of L over K, ν(x) and ν(y) are positive and the residue of x in K(Vp) is a transcendence basis of K(Vp) over K. Let w be a primitive element of L over K(x, y) such that ν(w) > 0. Let T = K[x, y, w] ⊂ V . Let f(x, y, z) be an irreducible element in the polynomial ring K[x, y, z] such that (8.9) K[x, y, z]/(f) =∼ K[x, y, w], where z + (f) is mapped to w. 146 8. Local Uniformization and Resolution of Surfaces

By our construction, the quotient ﬁeld of T is L and trdegK (T/p ∩ T ) = 1. Thus p ∩ T is a prime ideal of a curve on the surface spec(T ).

Set R0 = RmV ∩T . Let C0 be the curve in spec(T ) with ideal p ∩ R0 in R0. Corollary 4.4 implies there exists a proper birational morphism π : X → spec(T ) which is a sequence of blow-ups of points such that the strict transform C0 of C0 on X is non-singular. Let X1 → X be the blow-up of C0. Since X1 → spec(T ) is proper, there exists a unique point a1 ∈ X1 such that V dominates R1 = OX1,a1 . R1 is a quotient of a 3-dimensional regular local ring (as explained in Lemma 5.4). Let C1 be the curve in X1 with ideal p ∩ R1 in R1.(R1)p∩R1 dominates Rp∩R and (R1)p∩R1 is a local ring of a point on the blow-up of the maximal ideal of Rp∩R. We can iterate this procedure to construct a sequence of local rings (with quotient ﬁeld L)

R0 → R1 → · · · → Ri → · · · such that V dominates Ri for all i, Ri is a quotient of a regular local ring of dimension 3 and (Ri+1)p∩Ri+1 is a local ring of the blow-up of the maximal ideal of (Ri)p∩Ri for all i. Furthermore, each Ri is a localization of a quotient of a polynomial ring in 3 variables.

Since (R0)p∩R0 can be considered as a local ring of a point on a plane curve over the ﬁeld K(x) (as is explained in the proof of Lemma 5.10), by

Theorem 3.15 (or Corollary 4.4) there exists an i such that (Ri)p∩Ri is a regular local ring. After replacing the T in (8.9) with a suitable aﬃne ring, we can now assume that Tp∩T is a regular local ring. If TmV ∩T is a regular local ring we −1 are done, so suppose TmV ∩T is not a regular local ring. Let p = π (p ∩ T ), where π : K[x, y, z] → T is the natural surjection. If g ∈ K[x, y, z] is a polynomial, we will denote ν(π(g)) by ν(g). Let I ⊂ K[x, y, z] be the ideal ∂f ∂f ∂f I = (f, ∂x , ∂y , ∂z ) deﬁning the singular locus of T . Since I 6⊂ p (as Tp∩T is ∂f ∂f ∂f a regular local ring), one of π( ∂x ), π( ∂y ), π( ∂z ) 6∈ p ∩ T . Thus ∂f ∂f ∂f min{ν( ), ν( ), ν( )} = (0, n) ∂x ∂y ∂z for some n ∈ N. Write

f = Fr + Fr+1 + ··· + Fm, where Fi is a form of degree i in x, y, z and r = ord(f). After possibly reindexing the x, y, z, we may assume that 0 < ν(x) ≤ ν(y) ≤ ν(z). Since K(V ) = K, there exist c, d ∈ k (which could be 0) such that

ν(y − cx) > ν(x) and ν(z − dx) > ν(x). 8.2. Local uniformization of algebraic function ﬁelds of surfaces 147

Thus we have a birational transformation K[x, y, z] → K[x1, y1, z1] (a quadratic transformation), where

y − cx z − dx x = x, y = , z = , 1 1 x 1 x with inverse

x = x1, y = x1(y1 + c), z = x1(z1 + d).

r In K[x1, y1, z1] we have f(x, y, z) = x1f1(x1, y1, z1), where

f1(x1, y1, z1) = Fr(1, y1 + c, z1 + d) + x1Fr+1(1, y1 + c, z1 + d) m−r + ··· + x1 Fm(1, y1 + c, z1 + d) is the strict transform of f. Let T1 = K[x1, y1, z1]/(f1). Then T → T1 is birational, T1 ⊂ V and m ∩ T1 = (x1, y1, z1). We calculate

∂f = xr−1(x ∂f1 − (y + c) ∂f1 − (z + d) ∂f + rf ), ∂x 1 1 ∂x1 1 ∂y1 1 ∂z1 1 ∂f = xr−1 ∂f1 , ∂y 1 ∂y1 ∂f = xr−1 ∂f1 . ∂z 1 ∂z1 Hence

∂f ∂f ∂f ∂f1 ∂f1 ∂f1 min{ν( ), ν( ), ν( )} ≥ (r − 1)ν(x1) + min{ν( ), ν( ), ν( )}. ∂x ∂y ∂z ∂x1 ∂y1 ∂z1

Since we have assumed that TmV ∩T is not regular, we have r > 1. We further have that ν(x1) > 0, so we conclude that

∂f1 ∂f1 ∂f1 min{ν( ), ν( ), ν( )} = (0, n1) < (0, n). ∂x1 ∂y1 ∂z1

Let r1 = ord(f1) ≤ r.(T1)mV ∩T1 is a regular local ring if and only if r1 = 1. Thus we may assume that r1 > 1.

We now apply to T1 a new quadratic transform, to get

∂f2 ∂f2 ∂f2 min{ν( ), ν( ), ν( )} = (0, n2) < (0, n1). ∂x2 ∂y2 ∂z2

Thus after a ﬁnite number of quadratic transforms T → Tl, we construct a

Tl such that (Tl)mV ∩Tl is a regular local ring which is dominated by V . Remark 8.5. Zariski proves local uniformization for arbitrary characteristic zero algebraic function ﬁelds in [87]. 148 8. Local Uniformization and Resolution of Surfaces

8.3. Resolving systems and the Zariski-Riemann manifold

Let L/K be an algebraic function field. A projective model of L is a projective K-variety whose function field is L. Let Σ(L) be the set of valuation rings of L. If X is a projective model of L, then there is a natural morphism πX : Σ(L) → X defined by πX (V ) = p if p is the (unique) point of X whose local ring is dominated by V . We will say that p is the center of V on X. There is a topology on Σ(L), where −1 a basis for the topology are the open sets πX (U) for an open set U of a projective model X of L. Zariski shows that Σ(L) is quasi-compact (every open cover has a finite subcover) in [89] and Section 17, Chapter VI [92]. Σ(L) is called the Zariski-Riemann manifold of L. Let N be a set of zero-dimensional valution rings of L. A resolving system of N is a finite collection {S1,...,Sr} of projective models of L such that for each V ∈ N, the center of V on at least one of the Si is a non-singular point. Theorem 8.6. Suppose that L is a two-dimensional algebraic function field over an algebraically closed field K of characteristic zero. Then there exists a resolving system for the set of all zero-dimensional valuation rings of L.

Proof. By local uniformization (Theorem 8.2), for each valuation W of L there exists a projective model XW of L such that the center of W on XW is non-singular. Let UW be an open neighborhood of the center of W in −1 XW which is non-singular, and let AW = πW (UW ), an open neighborhood of W in Σ(L). The set {AW | W ∈ Σ(L)} is an open cover of Σ(L). Since

Σ(L) is quasi-compact, there is a ﬁnite subcover {AW1 ,...,AWr } of Σ(L).

Then XW1 ,...,XWr is a resolving system for the set of all zero-dimensional valuation rings of L. Lemma 8.7. Suppose that L is a two-dimensional algebraic function ﬁeld over an algebraically closed ﬁeld K of characteristic zero, and

R0 ⊂ R1 ⊂ · · · ⊂ Ri ⊂ · · · is a sequence of distinct normal two-dimensional algebraic local rings of S L. Let Ω = Ri. If every one-dimensional valuation ring V of L which contains Ω intersects Ri in a height one prime for some i, then Ω is the valuation ring of a zero-dimensional valuation of L.

Proof. Ω is by construction integrally closed in L. Thus Ω is the intersection of the valuation rings of L containing Ω (cf. the corollary to Theorem 8, Section 5, Chapter VI [92]). Observe that Ω 6= L. We see this as follows. S If mi is the maximal ideal of Ri, then m = mi is a non-zero ideal of Ω. Suppose that 1 ∈ m. Then 1 ∈ mi for some i, which is impossible. 8.3. Resolving systems and the Zariski-Riemann manifold 149

We will show that Ω is an intersection of zero-dimensional valuation rings. To prove this, it suffices to show that if V is a one-dimensional valuation ring containing Ω, then there exists a zero-dimensional valuation ring W such that V is a localization of W and Ω ⊂ W . Since V is one- dimensional, V is a local Dedekind domain and is an algebraic local ring of L. Let K(V ) be the residue field of V . Let mV be the maximal ideal of V , and let pi = Ri ∩ mV . By assumption, there exists an index j such that i ≥ j implies that pi is a height one prime in Ri.(Ri)pi is a normal local ring which is dominated by V . Since both (Ri)pi and V are local Dedekind domains, and they have the common quotient field L, we have that (Ri)pi = V for i ≥ j (cf. Theorem 9, Section 5, Chapter VI [92]).

Thus the residue field Ki of (Ri)pi is K(V ) for i ≥ j. The one-dimensional algebraic local rings Si = Ri/pi for i ≥ j have quotient field K(V ), and Si+1 S dominates Si. Si is not K(V ) by the same argument we used to show that Ω is not L. Thus there exists a valuation ring V 0 of K(V ) which dominates S −1 0 Si. Let π : V → K(V ) be the residue map. Then W = π (V ) is a zero-dimensional valuation ring of V and W contains Ω. Since we have established that Ω is the intersection of zero-dimensional valuation rings, we need only show that Ω is not contained in two distinct zero-dimensional valuation rings. Suppose that Ω is contained in two distinct zero-dimensional valuations, V1 and V2. We will first construct a projective surface Y with function field L such that V1 and V2 dominate (unique) distinct points q1 and q2 of Y . Let x, y be a transcendence basis of L over K. Let ν1, ν2 be valuations of L whose respective valuation rings are V1 and V2. If ν1(x) and ν2(x) are both ≤ 0, 1 replace x with x . Suppose that ν1(x) ≥ 0 and ν2(x) < 0. Let c ∈ K be such ∼ that the residue of x in the residue field K(V1) = K of V1 is not c. After replacing x with x − c, we have that ν1(x) = 0 and ν2(x) < 0. We can now 1 replace x with x to get that ν1(x) ≥ 0 and ν2(x) ≥ 0. In this way, after possibly changing x and y, we can assume that x and y are contained in the valuation rings V1 and V2. Since V1 and V2 are distinct zero-dimensional valuation rings, there exist f ∈ V1 such that f 6∈ V2, and g ∈ V2 such that g 6∈ V1 (cf. Theorem 3, Section 3, Chapter VI [92]). As above, we can 1 assume that ν1(f) = 0, ν2(f) < 0, ν1(g) < 0 and ν2(g) = 0. Let a = f , 1 b = g , and let A be the integral closure of K[x, y, a, b] in L. By construction,

A ⊂ V1 ∩ V2. Let m1 = A ∩ mV1 , m2 = A ∩ mV2 . We have that a ∈ m2 but a 6∈ m1, and b ∈ m1 but b 6∈ m2. Thus m1 6= m2. We now let Y be the normalization of a projective closure of spec(A). Y has the desired properties.

Since Ri is essentially of finite type over K, it is a localization of a finitely generated K-algebra Bi which has quotient field L. Let Xi be the 150 8. Local Uniformization and Resolution of Surfaces

normalization of a projective closure of spec(Bi). Since Ri is integrally closed, the projective surface Xi has a point ai with associated local ring

OXi,ai = Ri. There is a natural birational map Ti : Xi → Y . We will now establish that Ti cannot be a morphism at ai. If Ti were a morphism at ai, there would be a unique point bi on Y such that Ri = OXi,ai dominates OYi,bi .

Since V1 and V2 both dominate OXi,ai , V1 and V2 both dominate OYi,bi , a contradiction. By Zariski’s main theorem (cf. Theorem V.5.2 [47]) there exists a one- dimensional valuation ring W of L such that W dominates OXi,ai and W dominates the local ring of the generic point of a curve on Y . Let Γi be the set of one-dimensional valuation rings W of K such that W dominates

OXi,ai and W dominates the local ring of the generic point of a curve on Y . Γi is a finite set, since the valuation rings W of Γi are in 1-1 correspondence −1 with the curves contained in π1 (ai), where π1 :ΓTi → Xi is the projection of the graph of Ti onto Xi. Since Ri+1 = OXi+1,ai+1 dominates Ri = OXi,ai , we have Γi+1 ⊂ Γi for all i. Since each Γi is a non-empty finite set, there exists a one-dimensional valuation ring W of L such that W dominates Ri for all i, a contradiction. Theorem 8.8. Suppose that L is a two-dimensional algebraic function field over an algebraically closed field K of characteristic zero, R is a two-dimensional normal algebraic local ring with quotient field L, and

R → R1 → R2 → · · · → Ri → · · · is a sequence of normal local rings such that Ri+1 is obtained from Ri by blowing up the maximal ideal, normalizing, and localizing at a maximal ideal S so that Ri+1 dominates Ri. Then Ω = Ri is a zero-dimensional valuation ring of L.

Proof. By Lemma 8.7, we are reduced to showing that there does not exist a one-dimensional valuation ring V of L such that Ω is contained in V and mV ∩ Ri is the maximal ideal for all i. Suppose that V is a one-dimensional valuation ring of L which contains Ω. Let ν be a valuation of L whose associated valuation ring is V . Since V is one-dimensional, there exists α ∈ V such that the residue of α in K(V ) is transcendental over K.

Let mi be the maximal ideal of Ri. Assume that mV ∩ R is the maximal ξ ∼ ideal m0 of R. Write α = η with ξ, η ∈ R. Then α 6∈ R, since R/m0 = K. Hence η ∈ m0, and also ξ ∈ m0 since ν(α) ≥ 0. Since R1 is a local ring of the normalization of the blow-up of m0, m0R1 is a principal ideal. In fact if 8.3. Resolving systems and the Zariski-Riemann manifold 151

ζ ∈ m0 is such that ν(ζ) = min{ν(f) | f ∈ m0}, we have that ζR1 = m0R1. ξ η Thus ξ1 = ζ ∈ R1 and η1 = ζ ∈ R1. If we assume that m ∩ R = m , we have that ξ , η ∈ m , α = ξ2 with V 1 1 1 1 1 η2 ξ2, η2 ∈ R2 and ν(ξ) > ν(ξ1) > 0, ν(ξ1) > ν(ξ2),

ν(η) > ν(η1) > 0, ν(η1) > ν(η2).

More generally, if we assume that mi = mV ∩ V for i = 1, . . . , n, we can ﬁnd elements ξ , η ∈ R for 1 ≤ i ≤ n such that α = ξi and i i i ηi

ν(ξ) > ν(ξ1) > ··· > ν(ξn) > 0,

ν(η) > ν(η1) > ··· > ν(ηn) > 0. Since the value group of V is Z, it follows that n ≤ min{ν(ξ), ν(η)}. Hence, for all i sufficiently large, mV ∩ Ri is not the maximal ideal. mV ∩ Ri must then be a height one prime in Ri, for otherwise we would have mV ∩Ri = (0), which would imply that V = L, against our assumption. Remark 8.9. The conclusions of this theorem are false in dimension 3 (Shannon [76]). Theorem 8.10. Suppose that S, S0 are normal projective surfaces over an algebraically closed field K of characteristic zero and T is a birational map 0 from S to S . Let S1 → S be the projective morphism defined by taking the normalization of the blow-up of the finitely many points of T , where T is 0 not a morphism. If the induced birational map T1 from S1 to S is not a morphism, let S2 → S1 be the normalization of the blow-up of the finitely many fundamental points of T1. We then iterate to produce a sequence of birational morphisms of surfaces

S ← S1 ← S2 ← · · · ← Si ← · · · . This sequence must be of ﬁnite length, so that the induced rational map 0 Si → S is a morphism for all i suﬃciently large.

Proof. Since Si is a normal surface, the set of points where Ti is not a 0 morphism is finite (cf. Lemma 5.1 [47]). Suppose that no Ti : Si → S is a morphism. Then there exists a sequence of points pi ∈ Si for i ∈ N such that pi+1 maps to pi and Ti is not a morphism at pi for all i. Let Ri = OSi,pi . The birational maps Ti give an identification of the function fields of the 0 S Si and S with a common field L. By Theorem 8.8, Ω = Ri is a zero- dimensional valuation ring. Hence there exists a point q ∈ S0 such that Ω dominates OS0,q. OS0,q is a localization of a K-algebra B = K[f1, . . . , fr] for some f1, . . . , fr ∈ L. Since B ⊂ Ω, there exists some i such that B ⊂ Ri. Since Ω dominates Ri, we necessarily have that Ri dominates OS0,q. Thus Ti 152 8. Local Uniformization and Resolution of Surfaces

is a morphism at pi, a contradiction to our assumption. Thus our sequence must be ﬁnite. Theorem 8.11. Suppose that S and S0 are projective surfaces over an algebraically closed ﬁeld K of characteristic zero which form a resolving system for a set of zero-dimensional valuations N. Then there exists a projective surface S∗ which is a resolving system for N.

Proof. By Theorem 8.10, there exists a birational morphism S → S obtained by a sequence of normalizations of the blow-ups of all points where the birational map to S0 is not defined. Now we construct a birational mor- 0 phism S → S0, applying the algorithm of Theorem 8.10 by only blowing up the points where the birational map to S is not defined and which are non- 0 singular points. The algorithm produces a birational map S → S which is 0 a morphism at all non-singular points of S . 0 Let S∗ be the graph of the birational map from S to S , with natural ∗ ∗ 0 ∗ projections π1 : S → S and π2 : S → S . We will show that S is a resolving system for N. Suppose that V ∈ N. Let p, p0, p0 and p∗ be the 0 ∗ centers of V on the respective projective surfaces S, S0, S and S . First suppose that p0 is a singular point. Then p must be non-singular, 0 as {S,S0} is a resolving system for N. The birational map from S to S is a morphism at p, and π1 is an isomorphism at p. Thus the center of V on S∗ is a non-singular point. Now suppose that p0 is a non-singular point. Then p0 is a non-singular 0 0 point of S and the birational map from S to S is a morphism at p0. Thus 0 ∗ π2 is an isomorphism at p , and the center of V on S is a non-singular point. Theorem 8.12. Suppose that L is a two-dimensional algebraic function field over an algebraically closed field of characteristic zero. Then there exists a non-singular projective surface S with function field L.

Proof. The theorem follows from Theorem 8.6 and Theorem 8.10, by induction on r applied to a resolving system {S1,...,Sr} for the set of zero- dimensional valuation rings of L. Remark 8.13. Zariski proves the generalization of Theorem 8.11 to dimension 3 in [90], and deduces resolution of singularities for characteristic zero 3-folds from his proof of local uniformization for characteristic zero algebraic function ﬁelds [87]. Abhyankar proves local uniformization in dimension three and characteristic p > 5, from which he deduces resolution of singularities for 3-folds of characteristic p > 5 [4]. 8.3. Resolving systems and the Zariski-Riemann manifold 153

Exercise 8.14. 1. Prove that all the birational extensions constructed in Section 8.2 are products of blow-ups of points and non-singular curves. 2. Identify where characteristic zero is used in the proofs of this chapter. All but one of the cases of Section 8.2 extend without great diﬃculty to characteristic p > 0. Some care is required to ensure that K(x, y) → L is separable. In [1], Abhyankar accomplishes this and gives a very diﬀerent proof in the remaining case to prove local uniformization in characteristic p > 0 for algebraic surfaces.

Chapter 9

Ramiﬁcation of Valuations and Simultaneous Resolution

Suppose that L is an algebraic function field over a field K, and trdegK L < ∞ is arbitrary. We will use the notation for valuation rings of Section 8.1. Suppose that R is a local ring contained in L. We will say that R is algebraic (or an algebraic local ring of L) if L is the quotient field of R and R is essentially of finite type over K. Let K(S) denote the residue field of a ring S containing K with a unique maximal ideal. We will also denote the maximal ideal of a ring S containing a unique maximal ideal by mS. Suppose that L → L∗ is a finite separable extension of algebraic function fields over K, and V ∗ is a valuation ring of L∗/K associated to a valuation ν∗ with value group Γ∗. Then the restriction ν = ν∗ | L of ν∗ to L is a valuation of L/K with valuation ring V = L ∩ V ∗. Let Γ be the value group of ν. There is a commutative diagram

L → L∗ ↑ ↑ V = L ∩ V ∗ → V ∗.

155 156 9. Ramiﬁcation of Valuations and Simultaneous Resolution

There are associated invariants (cf. page 53, Section 11, Chapter VI [92]) namely, the reduced ramification index of ν∗ relative to ν, e = [Γ∗ : Γ] < ∞, and the relative degree of ν∗ with respect to ν, f = [K(V ∗): K(V )]. The classical case is when V ∗ is an algebraic local Dedekind domain. In this case V is also an algebraic local Dedekind domain. If mV ∗ = (y) and ∗ mV = (x), then there is a unit γ ∈ V such that x = γye. Since we thus have ∗ ∼ Γ /Γ = Z/eZ, we see that e is the reduced ramification index. Still assuming that V ∗ is an algebraic local Dedekind domain, if we further assume that K = K(V ∗) is algebraically closed, and (e, char(K)) = 1, then we have that the induced homomorphism on completions with respect to the respective maximal ideals Vˆ → Vˆ ∗ is given by the natural inclusion of K algebras K[[x]] → K[[y]], √ where y = e γy and x = ye. We thus have a natural action of Γ∗/Γ on Vˆ ∗ (a generator multiplies y by a primitive e-th root of unity), and the ring of invariants by this action is ∗ Vˆ =∼ (Vˆ ∗)Γ /Γ. In this chapter we find analogs of these results for general valuation rings. The basic approach is to find algebraic local rings R of L and S of L∗ ∗ ∗ such that V dominates S (S is contained in V and mV ∗ ∩ S = mS) and V dominates R, and such that the ramification theory of V → V ∗ is captured in (9.1) R → S, and we have a theory for R → S comparable to that of the above analysis of V → V ∗ in the special case when V ∗ is an algebraic local Dedekind domain. This should be compared with Zariski’s Local Uniformization Theorem [87] (also Section 2.5 and Chapter 8 of this book). It is proven in [87] that if char(K) = 0 and V is a valuation ring of an algebraic function field L/K, then V dominates an algebraic regular local ring of L. The case when trdegK L = 2 is proven in Section 8.2. 9. Ramification of Valuations and Simultaneous Resolution 157

We first consider the following diagram: L → L∗ (9.2) ↑ S where L∗/L is a finite separable extension of algebraic function fields over L, and S is a normal algebraic local ring of L∗. We say that S lies above an algebraic local ring R of L if S is a localization at a maximal ideal of the integral closure of R in L∗. Abhyankar and Heinzer have given examples of diagrams (9.2) where there does not exist a local ring R of L such that S lies above R (cf. [35]). To construct an extension of the kind (9.1), we must at least be able to find an S dominated by V ∗ such that S lies over an algebraic local ring R of L. To do this, we consider the concept of a monoidal transform. Suppose that V is a valuation ring of the algebraic function field L/K, and suppose that R is an algebraic local ring of L such that V dominates R. Suppose that p ⊂ R is a regular prime; that is, R/p is a regular local ring. p If f ∈ p is an element of minimal value, then R[ f ] is contained in V , and p p if Q = mV ∩ R[ f ], then R1 = R[ f ]Q is an algebraic local ring of L which is dominated by V . We say that R → R1 is a monoidal transform along V . R1 is the local ring of the blow-up of spec(R) at the non-singular subscheme V (p). If R is a regular local ring, then R1 is a regular local ring. In this case there exists a regular system of parameters (x1, . . . , xn) in R such that if ht(p) = r, then R = R[ x2 , ··· , xr ] . 1 x1 x1 Q The main tool we use for our analysis is the Local Monomialization Theorem 2.18. With the notation of (9.1) and (9.2), consider a diagram L → L∗ ↑ ↑ (9.3) V = V ∗ ∩ L → V ∗ ↑ S∗ where S∗ is an algebraic local ring of L∗. The natural question to consider is local simultaneous resolution; that is, does there exist a diagram (constructed from (9.3)) V → V ∗ ↑ ↑ R → S ↑ S∗ 158 9. Ramification of Valuations and Simultaneous Resolution such that S∗ → S is a sequence of monoidal transforms, S is a regular local ring, and there exists a regular local ring R such that S lies above R?

∗ The answer to this question is no, even when trdegK (L ) = 2, as was shown by Abhyankar in [2]. However, it follows from a reﬁnement in Theo- rem 4.8 [35] (strong monomialization) of Theorem 2.18 that if K has char- ∗ ∗ acteristic zero, trdegK (L ) is arbitrary and V has rational rank 1, then local simultaneous resolution is true, since in this case (2.8) becomes

a11 x1 = δ1y1 , x2 = y2, ··· , xn = yn, and S lies over R0. The natural next question to consider is weak simultaneous local resolution; that is, does there exist a diagram (constructed from (9.3)) V → V ∗ ↑ ↑ R → S ↑ S∗ such that S∗ → S is a sequence of monoidal transforms, S is a regular local ring, and there exists a normal local ring R such that S lies above R?

This was conjectured by Abhyankar on page 144 of [8] (and is implicit in ∗ [1]). If trdegK (L ) = 2, the answer to this question is yes, as was shown by ∗ Abhyankar in [1], [3]. If trdegK (L ) is arbitrary, and K has characteristic zero, then the answer to this question is yes, as we prove in [29] and [35]. This is a simple corollary of local monomialization. In fact, weak simultaneous local resolution follows from the following theorem, which will be of use in our analysis of ramification. Theorem 9.1 (Theorem 4.2 [35]). Let K be a field of characteristic zero, L an algebraic function field over K, L∗ a finite algebraic extension of L, and ν∗ a valuation of L∗/K, with valuation ring V ∗. Suppose that S∗ is an algebraic local ring of L∗ which is dominated by ν∗, and R∗ is an algebraic local ring of L which is dominated by S∗. Then there exists a commutative diagram ∗ R0 → R → S ⊂ V (9.4) ↑ ↑ R∗ → S∗ ∗ ∗ ∗ where S → S and R → R0 are sequences of monoidal transforms along ν such that R0 → S have regular parameters of the form of the conclusions of Theorem 2.18, R is a normal algebraic local ring of L with toric singularities 9. Ramification of Valuations and Simultaneous Resolution 159

which is the localization of the blow-up of an ideal in R0, and the regular local ring S is the localization at a maximal ideal of the integral closure of R in L∗.

Proof. By resolution of singularities [52] (cf. Theorems 2.6 and 2.9 in [26]), we ﬁrst reduce to the case where R∗ and S∗ are regular, and then construct, by the Local Monomialization Theorem 2.18, a sequence of monoidal transforms along ν∗

∗ R0 → S ⊂ V (9.5) ↑ ↑ R∗ → S∗ so that R0 is a regular local ring with regular parameters (x1, . . . , xn), S is a regular local ring with regular parameters (y1, . . . , yn), and there are units δ1, . . . , δn in S, and a matrix of natural numbers A = (aij) with non-zero determinant d such that

ai1 ain xi = δiy1 ··· yn for 1 ≤ i ≤ n. After possibly reindexing the yi, we may assume that d > 0. Let (bij) be the adjoint matrix of A. Set n n Y bij Y bij d fi = xj = ( δj )yi j=1 j=1 for 1 ≤ i ≤ n. Let R be the integral closure of R0[f1, . . . , fn] in L, localized ∗ √ at the center of ν . Since mRS = mS, Zariski’s Main Theorem (10.9 in [4]) shows that R is a normal algebraic local ring of L such that S lies above R. We thus have a sequence of the form (9.4).

In the theory of resolution of singularities a modiﬁed version of the weak simultaneous resolution conjecture is important. This is called global weak simultaneous resolution. Suppose that f : X → Y is a proper, generically ﬁnite morphism of k-varieties. Does there exist a commutative diagram

f1 X1 → Y1 (9.6) ↓ ↓ f X → Y such that f1 is ﬁnite, X1 and Y1 are complete k-varieties with X1 non- singular, Y1 normal, and the vertical arrows are birational? 160 9. Ramiﬁcation of Valuations and Simultaneous Resolution

This question has been posed by Abhyankar (with the further conditions that Y1 → Y is a sequence of blow-ups of non-singular subvarieties and Y , X are projective) explicitly on page 144 of [8] and implicitly in the paper [2]. We answer this question in the negative, in an example constructed In Theorem 3.1 of [30]. The example is of a generically finite morphism X → Y of non-singular, projective surfaces, defined over an algebraically closed field K of characteristic not equal to 2. This example also gives a counterexample to the principle that a geometric statement that is true locally along all valuations should be true globally. We now state our main theorem on ramification of arbitrary valuations. Theorem 9.2 (Theorem 6.1 [35]). Suppose that char(K) = 0 and we are given a diagram of the form of (9.3) L → L∗ ↑ ↑ V = V ∗ ∩ L → V ∗ ↑ S∗. Let K be an algebraic closure of K(V ∗). Then there exists a regular algebraic local ring R1 of L such that if R0 is a regular algebraic local ring of L which contains R1 such that ∗ R0 → R → S ⊂ V ↑ S∗ is a diagram satisfying the conclusions of Theorem 9.1, then: n t n ∼ ∗ 1. There is an isomorphism Z /A Z = Γ /Γ given by ∗ ∗ (b1, . . . , bn) 7→ b1ν (y1) + ··· + bnν (yn). ∗ ˆ 2. Γ /Γ acts on S⊗ˆ K(S)K, and the invariant ring is ˆ ∼ ˆ Γ∗/Γ R⊗ˆ K(R)K = (S⊗ˆ K(S)K) . 3. The reduced ramification index is e = |Γ∗/Γ| = | Det(A)|. 4. The relative degree is f = [K(V ∗): K(V )] = [K(S): K(R)].

It is shown in [35] and [32] that we can realize the valuation rings V and V ∗ as directed unions of local rings of the form of R and S in the above 9. Ramiﬁcation of Valuations and Simultaneous Resolution 161 theorem, and the union of the valuation rings is a Galois extension with Galois group Γ∗/Γ We now give an overview of the proof of Theorem 9.2, referring to [35] for details. ν∗ is our valuation of L∗ with valuation ring V ∗, and ν = ν∗ | L is the restiction of ν∗ to K. In the statement of the theorem Γ∗ is the value group of ν∗ and Γ is the value group of ν. By assumption, we have regular parameters (x1, . . . , xn) in R0 and (y1, . . . , yn) in S which satisfy the equations n Y aij xi = δi yj for 1 ≤ i ≤ n j=1 of (2.8). We have the relations n X ∗ ν(xi) = aijν (yj) ∈ Γ j=1 for 1 ≤ i ≤ n. Thus there is a group homomorphism n t n ∗ Z /A Z → Γ /Γ deﬁned by 1. To show that this homomorphism is an isomorphism we need an extension of local monomialization (proven in Theorem 4.9 [35]). 3 is immediate from 1

R0 → S is monomial, and R0 → R is a weighted blow-up. Lemma 4.4 of ∗ ˆ [35] shows that if M is the quotient field of S⊗ˆ K(S)K and M is the quotient ˆ ∗ ∗ ∼ field of R⊗ˆ K(R)K, then M /M is Galois with Galois group Gal(M /M) = n n ∼ n t n ˆ n/A n ∼ ˆ Z /AZ = Z /A Z . Furthermore, (S⊗ˆ K(S)K)Z Z = R⊗ˆ K(R)K. ˆ We see that R ⊗K(R) K is a quotient singularity, by a group whose invariant factors are determined by Γ∗/Γ. This observation leads to the counterexample to global weak simultaneous resolution stated earlier in this section. This example is written up in detail in [30]. We will give a very brief outline of the construction. With the notation of (9.6), if a valuation ν ∗ ∗ of K(Y ) has at least two distinct extensions ν1 and ν2 to K(X) which have ∗ ∗ non-isomorphic quotients Γ1/Γ and Γ2/Γ, then this presents an obstruction to the existence of a map X1 → Y1 (with the notation of (9.6)) which is finite, and X1 is non-singular. If there were such a map, we would have points p1 and p2 on X1, whose local rings R1 and R2 are regular and are ∗ ∗ dominated by ν1 and ν2 respectively, and a point p on Y1 whose local ring R is dominated by p. If we choose our valuation ν carefully, we can ensure ˆ ˆ ∗ ˆ ˆ ∗ that R is a quotient of R1 by Γ1/Γ and R is also a quotient of R2 by Γ2/Γ. We can in fact choose the extended valutions so that we can conclude that not only are these quotient groups not isomorphic, but the invariant rings are not isomorphic, which is a contradiction. 162 9. Ramification of Valuations and Simultaneous Resolution

An interesting question is whether the conclusions of local monomialization (Theorem 2.18) hold if the ground field K has positive characteristic. ∗ This is certainly true if trdegK (L ) = 1. ∗ Now suppose that trdegK (L ) = 2 and K is algebraically closed of positive characteristic. Strong monomialization (a refinement in Theorem 4.8 [35] of Theorem 2.18) is true if the value group Γ∗ of V ∗ is a finitely generated group (Theorem 7.3 [35]) or if V ∗/V is defectless (Theorem 7.35 [35]). The defect is an invariant which is trivial in characteristic zero, but is a power of p in an extension of characteristic p function fields. Theorem 7.38 [35] gives an example showing that strong monomializa- ∗ tion fails if trdegK (L ) = 2 and char(K) > 0. Of course the value group is not finitely generated, and V ∗/V has a defect. This example does satisfy the less restrictive conclusions of local monomialization (Theorem 2.18). It follows from strong monomialization in characteristic zero (a refinement of Theorem 2.18) that local simultaneous resolution is true (for arbi- ∗ ∗ trary trdegK (L )) if K has characteristic zero and rational rank ν = 1. We ∗ consider this condition when trdegK (L ) = 2 and K is algebraically closed of positive characteristic. In Theorem 7.33 [35] it is shown that in many cases ∗ local simultaneous resolution does hold if trdegK (L ) = 2, char(K) > 0 and the value group is not finitely generated. For instance, local simultaneous resolution holds if Γ is not p-divisible. The example of Theorem 7.38 [35] discussed above does in fact satisfy local simultaneous resolution, although it does not satisfy strong monomialization. Appendix. Smoothness and Non-singularity II

In this appendix, we prove the theorems on the singular locus stated in Section 2.2, and prove theorems on upper semi-continuity of order needed in our proofs of resolution. The proofs in Section A.1 are based on Zariski’s original proofs in [85].

A.1. Proofs of the basic theorems

n Suppose that P ∈ AK = spec(K[x1, . . . , xn]) has height n − r. Let mP ⊂ n OA ,P be the ideal of P . The diﬀerential 1 d : O n → Ω n = O n dx ⊕ · · · ⊕ O n dx A A A 1 A n is deﬁned by ∂u ∂u d(u) = dx1 + ··· + dxn. ∂x1 ∂xn d induces a linear transformation of K(P )-vector spaces 2 1 d : m /(m ) → Ω n ⊗ K(P ). P P P A ,P

n Deﬁne D(P ) to be the image of dP . Since OA ,P is a regular local ring, 2 dimK(P ) mP /mP = n−r. Thus dimK(P ) D(P ) ≤ n−r, and dimK(P ) D(P ) = n − r if and only if 2 dP : mP /(mP ) → D(P ) is a K(P )-linear isomorphism.

n Theorem A.1. Suppose that P ∈ AK = spec(K[x1, . . . , xn]) is a closed point. Then dimK(P ) D(P ) = n if and only if K(P ) is separable over K.

163 164 Appendix. Smoothness and Non-singularity II

Proof. Suppose that P ∈ spec(K[x1, . . . , xn]) is a closed point. Let mP ⊂ n OA ,P be the ideal of P . Let αi be the image of xi in K(P ) for 1 ≤ i ≤ n. Let f1(z1) ∈ K[z1] be the minimal polynomial of α1 over K. We can then inductively deﬁne fi(z1, . . . , zi) in the polynomial ring K[z1, . . . , zi] so that fi(α1, . . . , αi−1, zi) is the minimal polynomial of αi over K(α1, . . . , αi−1) for 2 ≤ i ≤ n. Let I be the ideal

n (f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)) ⊂ OA ,P . ∼ By construction I ⊂ mP and K[x1, . . . , xn]/I = K(P ), so we have I = mP . Thus f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn) is a regular system of parameters in O n , and the images of f (x ), f (x , x ), . . . , f (x , . . . , x ) AK ,P 1 1 2 1 2 n 1 n 2 in mP /mP form a K(P )-basis. Thus dimK(P ) D(P ) = n if and only if dP (f1), . . . , dP (fn) are linearly independent over K(P ). This holds if and only if J(f; x) has rank n at P , which in turn holds if and only if the determinant n Y ∂fi |J(f; x)| = ∂x i=1 i is not zero in K(P ). This condition holds if and only if K(α1, . . . , αi) is separable over K(α1, . . . , αi−1) for 1 ≤ i ≤ n, which is true if and only if K(P ) is separable over K. n Corollary A.2. Suppose that P ∈ AK = spec(K[x1, . . . , xn]) is a closed point such that K(P ) is separable over K. Let mP ⊂ OAn,P be the ideal of P , and suppose that u1, . . . , un ∈ mP . Then u1, . . . , un is a regular system of parameters in O n if and only if J(u; x) has rank n at P . AK ,P

2 Proof. This follows since dP : mP /mP → D(P ) is an isomorphism if and only if K(P ) is separable over K. Lemma A.3. Suppose that A is a regular local ring of dimension b with maximal ideal m, and p ⊂ A is an equidimensional ideal such that dim(A/p) = a. Then 2 2 dimA/m(p + m /m ) ≤ b − a, 2 2 and dimA/m(p + m /m ) = b − a if and only if A/p is a regular local ring.

Proof. Let A0 = A/p, m0 = mA0. k = A/m =∼ A0/m0. There is a short exact sequence of k-vector spaces (A.1) 0 → p/p∩m2 =∼ p+m2/m2 → m/m2 → m0/(m0)2 = m/(p+m2) → 0. The lemma now follows, since 0 0 2 dimk m /(m ) ≥ a 0 0 0 2 and A is regular if and only if dimk m /(m ) = a (Corollary 11.5 [13]). A.1. Proofs of the basic theorems 165

We now give two related results, Lemma A.4 and Corollary A.5. Lemma A.4. Suppose that A is a regular local ring of dimension n with maximal ideal m, p ⊂ A is a prime ideal such that A/p is regular, and dim(A/p) = n − r. Then there exist regular parameters u1, . . . , un in A such that p = (u1, . . . , ur) and ur+1, . . . , un map to regular parameters in A/p. Proof. Consider the exact sequence (A.1). Since A and A0 are regular, there exist regular parameters u1, . . . , un in A such that ur+1, . . . , un map 0 to regular parameters in A , and u1, . . . , ur are in p.(u1, . . . , ur) is thus a prime ideal of height r contained in p. Thus p = (u1, . . . , ur). Corollary A.5. Suppose that Y is a subvariety of dimension t of a variety X of dimension n. Suppose that q ∈ Y is a non-singular closed point of both Y and X. Then there exist regular parameters u1, . . . , un in OX,q such that IY,q = (u1, . . . , un−t), and un−t+1, . . . , un map to regular parameters in OY,q. If vn−t+1, . . . , vn are regular parameters in OY,q, there exist regular parameters u1, . . . , un as above such that ui maps to vi for n −t+ 1 ≤ i ≤ n.

Lemma A.6. Let I = (f1, . . . , fm) ⊂ K[x1, . . . , xn] be an ideal. Suppose that P ∈ V (I) and J(f; x) has rank n at P . Then K(P ) is separable algebraic over K, and n of the polynomials f1, . . . , fm form a system of regular parameters in O n . AK ,P

Proof. Let m be the ideal of P in K[x1, . . . , xn]. After possibly reindexing the fi, we may assume that |J(f1, . . . , fn; x)| 6∈ m. By the Nullstellensatz, m is the intersection of all maximal ideals n of K[x1, . . . , xn] containing m. n Thus there exists a closed point Q ∈ AK with maximal ideal n containing m such that |J(f1, . . . , fn; x)| 6∈ n, so that J(f1, . . . , fn; x) has rank n at n. We thus have dimK(Q) D(Q) = n, so that K(Q) is separable over K and n (f1, . . . , fn) is a regular system of parameters in OA ,Q by Theorem A.1 and Corollary A.2. Since I ⊂ m ⊂ n, we have P = Q. n Lemma A.7. Suppose that P ∈ AK = spec(K[x1, . . . , xn]) with ideal

m ⊂ K[x1, . . . , xn]

n is such that m has height n − r and t1, . . . , tn−r ∈ mm ⊂ OA ,P . Let αi be the image of xi in K(P ) for 1 ≤ i ≤ r. Then the determinant |J(t1, . . . , tn−r; xr+1, . . . , xn)| is non-zero at P if and only if α1, . . . , αr is a separating transcendence basis of K(P ) over K and t1, . . . , tn−r is a reg- n ular system of parameters in OA ,P .

Proof. Suppose that |J(t1, . . . , tn−r; xr+1, . . . , xn)| is non-zero at P . There exist s0, s1, . . . , sn−r ∈ K[x1, . . . , xn] such that si ∈ K[x1, . . . , xn] for all i, s 6∈ m and si = t for 1 ≤ i ≤ n − r. Then 0 s0 i |J(s1, . . . , sn−r; xr+1, . . . , xn)| 166 Appendix. Smoothness and Non-singularity II

∗ is non-zero at P . Let K = K(α1, . . . , αr), let n be the kernel of ∗ K [xr+1, . . . , xn] → K(P ), ∗ and let si for 1 ≤ i ≤ n − r be the image of si under ∗ K[x1, . . . , xn] → K [xr+1, . . . , xn]. n−r ∗ n is the ideal of a point Q ∈ AK∗ . We have K(P ) = K (Q), and at Q, ∗ ∗ |J(s1, . . . , sn−r; xr+1, . . . , xn)|= 6 0. ∗ ∗ ∗ ∗ By Lemma A.6, K (Q) is separable algebraic over K and (s1, . . . , sn−r) ∗ is a regular system of parameters in O n−r = K [xr+1, . . . , xn]n. Since AK∗ ,Q trdegK K(P ) = r, α1, . . . , αr is a separating transcendence basis of K(P ) ∗ over K. Thus K ⊂ K[x1, . . . , xn]m and ∗ K[x1, . . . , xn]m = K [xr+1, . . . , xn]n, so that t1, . . . , tn−r is a regular system of parameters in

O n = K[x , . . . , x ] . AK ,P 1 n m

Now suppose that α1, . . . , αr is a separating transcendence basis of K(P ) over K and t , . . . , t is a regular system of parameters in O n = 1 n−r AK ,P ∗ K[x1, . . . , xn]m. Let K = K(α1, . . . , αr), and let n be the kernel of ∗ K [xr+1, . . . , xn] → K(P ). n−r ∗ ∗ n is the ideal of a point Q ∈ AK∗ . Then K (Q) = K(P ) and K (Q) is separable over K∗. ∗ We have a natural inclusion K ⊂ K[x1, . . . , xn]m. Thus ∗ K[x1, . . . , xn]m = K [xr+1, . . . , xn]n, ∗ and t1, . . . , tn−r is a regular system of parameters in K [xr+1, . . . , xn]n. By Corollary A.2, |J(t1, . . . , tn−r; xr+1, . . . , xn)| 6= 0 at Q, and thus it is also 6= 0 at P . n Theorem A.8. Suppose that P ∈ AK = spec(K[x1, . . . , xn]) and t1, . . . , tn−r n are regular parameters in OA ,P . Then rank(J(t; x)) = n − r at P if and only if K(P ) is separably generated over K.

Proof. Let αi, 1 ≤ i ≤ n, be the images of xi in K(P ), and let m be the ideal of P in K[x1, . . . , xn]. Suppose that rank(J(t; x)) = n − r at P . After possibly reindexing the xi and tj, we have |J(t1, . . . , tn−r; xr+1, . . . , xn)|= 6 0 at P . Now K(P ) is separably generated over K by Lemma A.7.

Suppose that K(P ) is separably generated over K. Let ζ1, . . . , ζr be a separating transcendence basis of K(P ) over K. There exists Ψj(x) ∈ Ψj (α) K[x , . . . , x ] for 0 ≤ j ≤ r with Ψ (x) 6∈ m such that ζj = . Let 1 n 0 Ψ0(α) A.1. Proofs of the basic theorems 167

n be the kernel of the K-algebra homomorphism K[x1, . . . , xn+r] → K(P ) deﬁned by xi 7→ αi for 1 ≤ i ≤ n, xi 7→ ζi−n if n < i. n is the ideal of n+r a point Q ∈ AK . Let Φ0(x), Φ1(x),..., Φn−r(x) ∈ K[x1, . . . , xn] be such that Φ0 6∈ m and Φi(x) ti = for 1 ≤ i ≤ n − r. Φ0(x) Let Φn−r+j(x) = xn+jΨ0(x) − Ψj(x) for 1 ≤ j ≤ r. Let I ⊂ K[x1, . . . , xn+r] be the ideal generated by Φ1,..., Φn; thus I ⊂ n. We will show that Φ1,..., Φn are in fact a regular system of parameters in

O n+r = K[x1, . . . , xn+r]n. AK ,Q It suﬃces to show that if F (x) ∈ n, then there exists

A(x) ∈ K[x1, . . . , xn+r] − n such that AF ∈ I. Given such an F , we have an expansion r s X Ψ0(x) F (x) = Bj(x)Φn−r+j(x) + G(x1, x2, . . . , xn), j=1 where Bj(x) ∈ K[x1, . . . , xn+r] and s is a non-negative integer. Now F ∈ n implies G ∈ m. Thus there exists an expansion n−r X G = fiti i=1 with fi ∈ K[x1, . . . , xn]m, and there exist h ∈ K[x1, . . . , xn] − {m} and Ci ∈ K[x1, . . . , xn] such that r X hG = CiΦi(x) i=1 s and hΨ0(x) F (x) ∈ I.

Since Φ1,..., Φn is a regular system of parameters in K[x1, . . . , xn+r]n and ζ1, . . . , ζr is a separating transcendence basis of K(Q) over K, it follows that |J(Φ1,..., Φn; x1, . . . , xn)| is non-zero at Q by Lemma A.7. Thus J(Φ1,..., Φn−r; x1, . . . , xn) is of rank n − r at P , and by Lemma A.7, α1, . . . , αn must contain a separating transcendence basis of K(P ) over K. Proof that smoothness at a point is well deﬁned

Suppose that X is a variety of dimension s over a field K and P ∈ X. In Definition 2.6 we gave a definition for smoothness of X over K at P which 168 Appendix. Smoothness and Non-singularity II depended on choices of an affine neighborhood U = spec(R) of P and a ∼ presentation R = K[x1, . . . , xn]/I with I = (f1, . . . , fm). We now show that this definition is well defined, so that it is independent of all choices U, f and x. 1 Consider the sheaf of differentials ΩX/K . We have, by the “second fundamental exact sequence” (Theorem 25.2 [66]), a presentation

J(f;x) (A.2) (K[x , . . . , x ])m → Ω1 ⊗ R → Ω1 → 0. 1 n K[x1,...,xn]/K R/K

1 Tensoring (A.2) with K(P ), we see that ΩR/K ⊗ K(P ) has rank s if and only if J(f; x) has rank n − s at P .

Proof of Theorem 2.8

Let s = dim X. There exists an aﬃne neighborhood U = spec(R) of P in X such that R = K[x1, . . . , xn]/I, I = (f1, . . . , fm). Let mP ∈ n spec(K[x1, . . . , xn]) be the ideal of P in AK . Let

t = dim(K[x1, . . . , xn]/mP ).

Suppose that K(P ) is separably generated over K and P is a non- 2 singular point of X. By Theorem A.8, dP : mP /mP → D(P ) is an isomor- n phism. By Lemma A.3, dimK(P ) dP (I) = n−s (take A = OA ,P , p = ImP in the statement of the lemma, so that b = n−t, a = s−t), which is equivalent to the condition that J(f; x) has rank n − s at P . Thus X is smooth over K at P . Suppose that X is smooth at P , so that J(f; x) has rank n − s at P . 2 2 Then dimK(P ) dP (I) = n − s, so that dimK(P )(I + mP )/mP ≥ n − s. Thus P is a non-singular point of X by Lemma A.3. Remark A.9. With the notation of Theorem 2.8 and it’s proof, for any point P ∈ V (I), we have 2 2 dimK(P )(I + mP )/mP ≤ n − s by Lemma A.3, so that J(f; x) has rank ≤ n − s at P .

Proof of Theorem 2.7

It suffices to prove that the locus of smooth points of X lying on an open affine subset U = spec(R) of X of the form of Definition 2.6 is open. Let A = In−s(J(f; x)). Then P ∈ U − V (A) if and only if J(f; x) has rank A.2. Non-singularity and uniformizing parameters 169 greater than or equal to n−s at P , which in turn holds if and only if J(f; x) has rank n − s at P by Remark A.9.

Proof of Theorem 2.10 when K is perfect

Suppose that K is perfect. The openness of the set of non-singular points of X follows from Theorem 2.7 and Corollary 2.9. Let η ∈ X be the generic point of an irreducible component of X. Then OX,η is a ﬁeld which is a regular local ring. Thus η is a non-singular point. We conclude that the non-singular points of X are dense in X.

A.2. Non-singularity and uniformizing parameters

Theorem A.10. Suppose that X is a variety of dimension r over a ﬁeld K, and P ∈ X is a closed point such that X is non-singular at P and K(P ) is separable over K.

1. Suppose that y1, . . . , yr are regular parameters in OX,P . Then 1 ΩX/K,P = dy1OX,P ⊕ · · · ⊕ dyrOX,P .

2. Let m be the ideal of P in OX,P . Suppose that f1, . . . , fr ∈ m ⊂ 1 OX,P and df1, . . . , dfr generate ΩX/K,P as an OX,P module. Then f1, . . . , fr are regular parameters in OX,P .

Proof. We can restrict to an aﬃne neighborhood of X, and assume that

X = spec(K[x1, . . . , xn]/I).

The conclusions of part 1 of the theorem follow from Theorem A.8 when n X = AK . In the general case of 1, by Corollary A.5 there exist regular parameters (y1,..., yn) in A = K[x1, . . . , xn]m (where m is the ideal of P in spec(K[x1, . . . , xn]) such that IA = (yr+1,..., yn) and (y1,..., yr) map to the regular parameters (y1, . . . , yr) in B = A/IA. By the second fundamental exact sequence (Theorem 25.2 [66]) we have an exact sequence of B-modules 2 1 1 IA/I A → ΩA/K ⊗A B → ΩB/K → 0. This sequence is thus a split short exact sequence of free B-modules, with 1 ∼ 2 ∼ 1 ΩB/K = dy1B ⊕· · ·⊕dyrB, since IA/I A = yr+1B ⊕· · ·⊕ynB and ΩA/K ⊗A ∼ B = dy1B ⊕ · · · ⊕ dynB. Suppose that the assumptions of 2 hold. Let R = OX,P . Let y1, . . . , yr be regular parameters in R. Then dy1, . . . , dyr is a basis of the R-module 170 Appendix. Smoothness and Non-singularity II

1 ΩX/K,P by part 1 of this theorem. There are aij ∈ R such that

r X fi = aijyj, 1 ≤ j ≤ r. j=1 We have expressions

r r X X dfi = aijdyj + ( daijyj). j=1 j=1 Pr Let wi = j=1 aijdyj for 1 ≤ i ≤ r. By Nakayama’s Lemma we have that 1 the wi generate ΩX/K,P as an R-module. Thus |(aij)| 6∈ m and (aij) is invertible over R, so that f1, . . . , fr generate m. Lemma A.11. Suppose that X is a variety of dimension r over a perfect ﬁeld K. Suppose that P ∈ X is a closed point such that X is non-singular at P , and y1, . . . , yr are regular parameters in OX,P . Then there exists an aﬃne neighborhood U = spec(R) of P in X such that y1, . . . , yr ∈ R, the r natural inclusion S = K[y1, . . . , yr] → R induces a morphism π : U → AK such that

1 (A.3) ΩR/K = dy1R ⊕ · · · ⊕ dyrR,

1 ∂ ∂ DerK (R,R) = HomR(ΩR/K ,R) = R ⊕ · · · ⊕ R ∂y1 ∂yr (with ∂ (dy ) = δ ), and the following condition holds. Suppose that a ∈ U ∂yi j ij is a closed point, b = π(a), and (x , . . . , x ) are regular parameters in O r . 1 r AK ,b Then (x1, . . . , xr) are regular parameters in OX,a.

y1, . . . , yr are called uniformizing parameters on U.

Proof. By Theorem A.10, there exists an aﬃne neighborhood U = spec(R) 1 of P in X such that y1, . . . , yr ∈ R and dy1, . . . , dyr is a free basis of ΩR/K . Consider the natural inclusion S = K[y1, . . . , yr] → R, with induced morphism π : spec(R) → spec(S). Suppose that a ∈ U is a closed point with maximal ideal m in R, and b = π(a), with maximal ideal n in S. Suppose that (x1, . . . , xr) are regular parameters in B = Sn. Let A = Rm. By con- 1 1 struction, ΩS/K is generated by dy1, . . . , dyr as an S-module. Thus ΩA/K is 1 generated by ΩB/K as an A-module. By 1 of Theorem A.10, dx1, . . . , dxr 1 1 generate ΩB/K as a B-module, and hence they generate ΩA/K as an A- module. By 2 of Theorem A.10, x1, . . . , xr is a regular system of parameters in A. A.3. Higher derivations 171

A.3. Higher derivations

Suppose that K is a ﬁeld and A is a K-algebra. A higher derivation of A over K of length m is a sequence

D = (D0,D1,...,Dm) if m < ∞, and a sequence

D = (D0,D1,...) if m = ∞, of K-linear maps Di : A → A such that D0 = id and i X (A.4) Di(xy) = Dj(x)Dj−i(y) j=1 for 1 ≤ i ≤ m and x, y ∈ A.

Let t be an indeterminate. D = (D0,D1,...) is a higher derivation of A over K (of length ∞) precisely when the map Et : A → A[[t]] deﬁned by P∞ i Et(f) = i=0 Di(f)t is a K-algebra homomorphism with D0(f) = f, and D = (D0,D1,...,Dm) is a higher derivation of A over K of length m pre- m+1 Pm i cisely when the map Et : A → A[t]/(t ) deﬁned by Et(f) = i=0 Di(f)t is a K-algebra homomorphism with D0(f) = f.

Example A.12. Suppose that A = K[y1, . . . , yr] is a polynomial ring over r a field K. For 1 ≤ i ≤ r, let j be the vector in N with a 1 in the j- P i1 ir th coefficient and 0 everywhere else. For f = ai1,...,ir y1 ··· yr ∈ A and m ∈ N, define X ij i −m D∗ (f) = a yi1 ··· y j ··· yir . mj i1,...,ir m 1 j r Then D∗ = (D∗,D∗ ,D∗ ,...) is a higher derivation of A over K of length j 0 j 2j ∗ ∞ for 1 ≤ j ≤ r. Observe that the Dj are iterative; that is, i + k D ◦ D = D ij kj i (i+k)j for all i, k. Define ∗ ∗ ∗ ∗ Di1,...,ir = Dirr ◦ Dir−1r−1 ◦ · · · ◦ Di11 for (i , . . . , i ) ∈ r. The D∗ commute with each other. 1 r N i1,...,ir The D∗ are called Hasse-Schmidt derivations [56]. If K has charac- i1,...,ir teristic zero, then

i1+···+ir ∗ 1 ∂ Di ,...,i = . 1 r i !i ! ··· i ! i1 ir 1 2 r ∂y1 ··· ∂yr 172 Appendix. Smoothness and Non-singularity II

Lemma A.13. Suppose that X is a variety of dimension r over a perfect field K. Suppose that P ∈ X is a closed point such that X is non-singular at P , y1, . . . , yr are regular parameters in OX,P , and U = spec(R) is an affine neighborhood of P in X such that the conclusions of Lemma A.11 hold. Then there are differential operators Di1,...,ir on R, which are uniquely determined on R by the differential operators D∗ on S of Example A.12, i1,...,ir such that if f ∈ OX,P and

X i1 ir f = ai1,...,ir y1 ··· yr ∼ in OˆX,P = K(P )[[y1, . . . , yr]], where we identify K(P ) with the coeﬃcient ˆ ﬁeld of OX,P containing K, then ai1,...,ir = Di1,...,ir (f)(P ) is the residue of

Di1,...,ir (f) in K(P ) for all indices i1, . . . , ir.

Proof. If K has characteristic zero, we can take 1 ∂i1+···+ir (A.5) Di ,...,i = . 1 r i ! ··· i ! i1 ir 1 r ∂y1 ··· ∂yr Suppose that K has characteristic p ≥ 0. Consider the inclusion of K-algebras S = K[y1, . . . , yr] → R of Lemma A.11, with induced morphism π : U = spec(R) → V = spec(S). ∗ Let Dj for 1 ≤ j ≤ r be the higher order derivation of S over K defined in 1 Example A.12. We have ΩR/S = 0 by 1 of Theorem A.10, (A.3) and the “first fundamental exact sequence” (Theorem 25.1 [66]). Thus π is non-ramified by Corollary IV.17.4.2 [45]. By Theorem IV.18.10.1 [45] and Proposition IV.6.15.6 [45], π is etale. ∗ We will now prove that Dj extends uniquely to a higher derivation Dj of ∗ ∗ ∗ R over K for 1 ≤ j ≤ r. It suffices to prove that (D0,Dj ,...,Dmj ) extends ∗ to a higher derivation of R over K for all m. The extension of D0 = id to D0 = id is immediate. Suppose that an extension (D0,D1,...,Dm) of ∗ ∗ ∗ (D0,Dj ,...,Dmj ) has been constructed. Then we have a commutative diagram of K-algebra homomorphisms g R → R[t]/(tm+1) ↑ ↑ f S → R[t]/(tm+2) where m X j g(a) = Dij (a)t , i=0 A.3. Higher derivations 173

m+1 X ∗ j f(a) = Dij (a)t . i=0 Since (tm+1)R[t]/(tm+2) is an ideal of square zero, and S → R is formally etale (Deﬁnitions IV.17.3.1 and 17.1.1 [45]), there exists a unique K-algebra homomorphism h making a commutative diagram g R → R[t]/(tm+1) h ↑ & ↑ f S → R[t]/(tm+2) Thus there is a unique extension of (D∗,D∗ ,...,D∗ ) to a higher 0 j (m+1)j derivation of R.

Set Di1,...,ir = Dirr ◦ · · · ◦ Di11 . Let A = OX,P . Since K(P ) is separable over K, and by Exercise 3.9 of Section 3.2, we can identify the residue field K(P ) of A with the coefficient field of Aˆ which contains K. Thus we have natural inclusions ∼ K[y1, . . . , yr] ⊂ A ⊂ Aˆ = K(P )[[y1, . . . , yr]]. ˆ We will now show that Dj extends uniquely to a higher derivation of A over K for 1 ≤ j ≤ r. Let m be the maximal ideal of A. Let t be an indeterminate. Then Aˆ[[t]] is a complete local ring with maximal ideal n = mAˆ[[t]] + tAˆ[[t]]. ˆ P∞ n Define Et : A → A[[t]] by Et(f) = n=0 Dnj (f)t . Et is a K-algebra homomorphism, since Dj is a higher derivation. Et is continuous in the a a m-adic topology, as Et(m ) ⊂ n for all a. Thus Et extends uniquely to ˆ ˆ a K-algebra homomorphsim A → A[[t]]. Thus Dj extends uniquely to a higher derivation of Aˆ over K. ˆ We can thus extend Di1,...,ir uniquely to A. Since K(P ) is algebraic over K, and D extends D∗ , it follows that D acts on Aˆ as stated i1,...,ir i1,...,ir i1,...,ir in the theorem. Remark A.14. Suppose that the assumptions of Lemma A.13 hold, and that K is an algebraically closed field. Then the following properties hold. 1. Suppose that Q ∈ U is a closed point and a is the corresponding maximal ideal in R. Then a = (y1,..., yr), where yi = yi − αi with ˆ ˆ αi = yi(Q) ∈ K for 1 ≤ i ≤ r, and K[[y1,..., yr]] = Ra = OU,Q.

2. The diﬀerential operators Di1,...,in on R are such that if f ∈ R, and a is a maximal ideal in R, then, with the notation of 1,

X i1 ir f = ai1,...,ir y1 ··· yr 174 Appendix. Smoothness and Non-singularity II

ˆ in K[[y1,..., yr]] = Ra, where ai1,...,ir = Di1,...,ir (f)(Q) is the

residue of Di1,...,ir (f) in K(Q) for all indices i1, . . . , ir.

Proof. The ideal of π(Q) in S is (y1 −α1, . . . , yr −αr) with αi = yi(Q) ∈ K. Let yi = yi − αi for 1 ≤ i ≤ r. We have a = (y1,..., yr) by Theorem A.10 2, and 1 follows. The derivations ∂ ,..., ∂ are exactly the derivations ∂y1 ∂yr ∂ ,..., ∂ on S of Example A.12, so they extend to the differential op- ∂y1 ∂yr erators D∗ on S of Example A.12. Now 2 follows from Lemma A.13 i1,...,ir applied to Q and y1,..., yr. Lemma A.15. Let notation be as in Lemmas A.11 and A.13. In particular, we have a fixed closed point P ∈ U = spec(R) and differential operators

Di1,...,ir on R. Let I ⊂ R be an ideal. Deﬁne an ideal Jm,U ⊂ R by

Jm,U = {Di1,...,ir (f) | f ∈ I and i1 + ··· + ir < m}.

Suppose that Q ∈ U is a closed point, with corresponding ideal mQ in R, and suppose that (z1, . . . , zr) is a regular system of parameters in mQ.

Let Di1,...,ir be the diﬀerential operators constructed on OW,Q by the construction of Lemmas A.11 and A.13. Then

(Jm,U )mQ = {Di1,...,ir (f) | f ∈ I and i1 + ··· + ir < m}.

Proof. When K has characteristic zero, this follows directly from differen- tiation and (A.5). Assume that K has positive characteristic. By Lemma A.13 and Theo- rem IV.16.11.2 [45], U is differentiably smooth over spec(K) and {Di1,...,ir | i1, . . . , ir < m} is an R-basis of the differential operators of order less than 1 m on R. Since {dz1, . . . , dzr} is a basis of Ω (by Lemma A.11), RmQ /K

{Di1,...,ir | i1, . . . , ir < m} is an RmQ -basis of the diﬀerential operators of order less than m on RmQ by Theorem IV.16.11.2 [45]. The lemma follows.

A.4. Upper semi-continuity of νq(I) Lemma A.16. Suppose that W is a non-singular variety over a perfect 0 0 0 ﬁeld K and K is an algebraic ﬁeld extension of K. Let W = W ×K K , 0 0 with projection π1 : W → W . Then W is a non-singular variety over K0. Suppose that q ∈ W is a closed point. Then there exists a closed point 0 0 0 q ∈ W such that π1(q ) = q. Furthermore,

1. Suppose that (f1, . . . , fr) is a regular system of parameters in OW,q. Then (f1, . . . , fr) is a regular system of parameters in OW 0,q0 . A.4. Upper semi-continuity of νq(I) 175

2. If I ⊂ OW,q is an ideal, then ˆ (IOW 0,q0 ) ∩ OW,q = I.

Proof. The fact that W 0 is non-singular follows from Theorem 2.8 and the 0 0 0 definition of smoothness. Let q ∈ W be a closed point such that π1(q ) = q. We first prove 1. Suppose that U = spec(R) is an affine neighborhood of q in W . There is a presentation R = K[x1, . . . , xn]/I. We further have −1 0 0 0 0 that π1 (U) = spec(R ), with R = K [x1, . . . , xn]/I(K [x1, . . . , xn]). Let m 0 0 be the ideal of q in K[x1, . . . , xn], and n the ideal of q in K [x1, . . . , xn]. By the exact sequence of Lemma A.3, there exists a regular system of parameters f 1,..., f n in K[x1, . . . , xn]m, such that Im = (f r+1,..., f n)m and f i maps to fi for 1 ≤ i ≤ r. By Corollary A.2, the determinant 0 |J(f 1,..., f n; x1, . . . , xn)| is non-zero at q, so it is non-zero at q . By Corol- 0 lary A.2, f 1,..., f n is a regular system of parameters in K [x1, . . . , xn]n. 0 Thus f1, . . . , fr is a regular system of parameters in Rn = OW 0,q0 . 0 We will now prove 2. Since n maps to a maximal ideal of OW,q ⊗K K ,

ˆ 0 ˆn OW 0,q0 = (OW,q ⊗K K ) 0 ˆ (ˆn denotes completion with respect to nOW,q ⊗K K ). Thus OW 0,q is faith- ˆ fully flat over OW,q, and (IOW 0,q0 )∩OW,q = I (by Theorem 7.5 (ii) [66]). Suppose that (R, m) is a regular local ring containing a field K of characteristic zero. Let K0 be the residue field of R. Suppose that J ⊂ R is an ideal. We define the order of J in R to be b νR(J) = max{b | J ⊂ m }. If J is a locally principal ideal, then the order of J is its multiplicity. How- ever, order and multiplicity are different in general.

Deﬁnition A.17. Suppose that q is a point on a variety W and J ⊂ OW is an ideal sheaf. We denote

νq(J) = νOW,q (JOW,q). If X ⊂ W is a subvariety, we denote

νq(X) = νq(IX ).

Remark A.18. Let notation be as in Lemma A.16 (except we allow K to be an arbitrary, not necessarily perfect ﬁeld). Observe that if q ∈ W and 0 0 0 q ∈ W are such that π(q ) = q and J ⊂ OW is an ideal sheaf, then

ˆ 0 0 ˆ 0 0 νOW,q (Jq) = ν ˆ (JqOW,q) = νO 0 0 (JqOW ,q ) = ν ˆ (JqOW ,q ). OW,q W ,q OW 0,q0 176 Appendix. Smoothness and Non-singularity II

Suppose that X is a noetherian topological space and I is a totally ordered set. f : X → I is said to be an upper semi-continuous function if for any α ∈ I the set {q ∈ X | f(q) ≥ α} is a closed subset of X. Suppose that X is a subvariety of a non-singular variety W . Deﬁne

νX : W → N by νX (q) = νq(X) for q ∈ W . The order νq(X) is defined in Definition A.17. Theorem A.19. Suppose that K is a perfect field, W is a non-singular variety over K and I is an ideal sheaf on W . Then

νq(I): W → N is an upper semi-continuous function.

Proof. Suppose that m ∈ N. Suppose that U = spec(R) is an open aﬃne subset of W such that the conclusions of Lemma A.11 hold on W . Set I = Γ(U, I).

With the notation of Lemmas A.11 and A.13, deﬁne an ideal Jm,U ⊂ R by

Jm,U = {Di1,...,ir (g) | g ∈ I and i1 + ··· ir < m}. By Lemma A.15, this deﬁnition is independent of the choice of y1, . . . , yr satisfying the conclusions of Lemma A.11, so our deﬁnition of Jm,U determines a sheaf of ideals Jm on W . We will show that

V (Jm) = {η ∈ W | νη(I) ≥ m}, from which upper semi-continuity of νq(I) follows. Suppose that η ∈ W is a point. Let Y be the closure of {η} in W , and suppose that P ∈ Y is a closed point such that Y is non-singular at P . By Corollary A.5, Lemma A.11 and Lemma A.15, there exists an aﬃne neighborhood U = spec(R) of P in W such that there are y1, . . . , yr ∈ R with the properties that y1 = ... = yt = 0 (with t ≤ r) are local equations of Y in U and y1, . . . , yr satisfy the conclusions of Lemma A.11.

For g ∈ IP , consider the expansion

X i1 ir g = ai1,...,ir y1 ··· yr ∼ in OˆW,P = K(P )[[y1, . . . , yr]], where we have identified K(P ) with the coefficient field of OˆW,P containing K and (with the notation of Lemma A.13)

ai1,...,ir = Di1,...,ir (g)(P ). A.4. Upper semi-continuity of νq(I) 177

Then η ∈ V (Jm) is equivalent to Di1,...,ir (g) ∈ (y1, . . . , yt) whenever g ∈ IP and i1 + ··· + ir < m, which is equivalent to

ai1,...,ir = Di1,...,ir (g)(P ) = 0 if g ∈ IP and i1 + ··· + it < m, which in turn holds if and only if ˆm m g ∈ IY,P ∩ OW,P = IY,P for all g ∈ IP , where the last equality is by Lemma A.16. Localizing OW,P at η, we see that νη(I) ≥ m if and only if η ∈ V (Jm). Definition A.20. Suppose that W is a non-singular variety over a perfect field K and X ⊂ W is a subvariety. For b ∈ N, define

Singb(X) = {q ∈ W | νq(W ) ≥ b}.

Singb(X) is a closed subset of W by Theorem A.19. Let

r = max{νX (q) | q ∈ W }.

Suppose that q ∈ Singr(X). A subvariety H of an aﬃne neighborhood U of q in W is called a hypersurface of maximal contact for X at q if

1. Singr(X) ∩ U ⊂ H, and 2. if πn π1 Wn → · · · → W1 → W

is a sequence of monoidal transforms such that for all i, πi is centered at a non-singular subvariety Yi ⊂ Singr(Xi), where Xi is the strict transform of X on Wi, then the strict transform Hn of H on Un = Wn ×W U is such that Singr(Xn) ∩ Un ⊂ Hn. With the above assumptions, a non-singular codimension one subvariety H of U = spec(OˆW,q) is called a formal hypersurface of maximal contact for X at q if 1 and 2 above hold for U = spec(OˆW,q). Remark A.21. Suppose that W is a non-singular variety over a perfect field K and X ⊂ W is a subvariety, q ∈ W . Let T = OˆW,q, and define ˆ Singb(OX,q) = {P ∈ spec(T ) | ν(IX,qTP ) ≥ b}. ˆ Then Singb(OX,q) is a closed set. If J ⊂ OW is a reduced ideal sheaf whose support is Singb(X), then JqT is a reduced ideal whose support is ˆ Singb(OX,q). Proof. Let U = spec(R) be an affine neighborhood of q in W satisfying the conclusions of Lemma A.13 (with p = q), and let notation be as in Lemma A.13 (with p = q). With the further notation of Theorem A.19 (and I = IX ) p we see that Γ(U, J) = Jb,U . JqT is a reduced ideal by Theorem 3.6. 178 Appendix. Smoothness and Non-singularity II

ˆ Suppose that P ∈ Singb(OX,q). Then νTP (g) ≥ b for all g ∈ IX,q. Thus g ∈ P (b) ⊂ T , and there exists h ∈ T − P such that gh ∈ P b. Now b−i1,...,ir induction in the formula (A.4) shows that Di1,...,ir (gh) ∈ P and (b−i1−···−ir) Di1,...,ir (g) ∈ P ⊂ P if i1 + ··· + ir < b. Thus JqT ⊂ P . Now suppose that P ∈ spec(T ) and JqT ⊂ P . Let S = OW,q. If Jq = P1 ∩· · ·∩Pm is a primary decomposition, then all primes Pi are minimal since Jq is reduced. By Theorem 3.6 there are primes Pij in T and a function σ(i) such that PiT = Pi1 ∩ · · · ∩ Piσ(i) is a minimal primary decomposition. T Thus JqTPij = PiTPij = PijTPij for all i, j. We have ij Pij ⊂ P , so Pij ⊂ P b b for some i, j. Moreover, IX,q ⊂ Pi SPi implies IX,q ⊂ PijTPij , which in turn b ˆ implies IX,q ⊂ P TPij . Thus νTP (IX,qTP ) ≥ b and P ∈ Singb(OX,q). Bibliography

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Index

Abhyankar, 12, 28, 42, 45, 130, 131, 152, intersection 153, 157, 158, 160 reduced set-theoretic, 62 Abramovich, 9 scheme-theoretic, 2, 62 Albanese, 55 isolated subgroup, 134 approximate manifold, 106 Atiyah, viii Jung, 55

Bierstone, 61 Karu, 9 Bravo, 61 Kawasaki, 11 Kuhlmann, 13 Cano, 13 Chevalley, 24 Levi, 55 Cohen, 22 Lipman, 130 completion of the square, 24 local uniformization, 12, 130, 138, 147, 156 Cossart, 131 curve, 2 Macaulayﬁcation, 11 Macdonald, viii de Jong, 61, 131 MacLane, 137 derivations Matsuki, 9 Hasse-Schmidt, 171 maximal contact higher, 171 formal curve, 28 formal hypersurface, 177 Eisenbud, viii hypersurface, 56, 67, 70, 131, 177 Encinas, 61 Milman, 61 exceptional divisor, 20, 39, 40 Moh, 61 exceptional locus, 39, 40 monoidal transform, 16, 39, 40 monomialization, 13, 157 Giraud, 70 morphism birational, 9 Harris, viii monomial, 14 Hartshorne, viii, 2 proper, 9 Hauser, 13, 61, 132 multiplicity, 3, 19, 25, 175 Heinzer, 13, 157 Hensel’s Lemma, 23 Nagata, 11 Hironaka, 2, 15, 61, 62, 130 Narasimhan, 131 hypersurface, 2 Newton, 1, 3, 6, 12

185 186 Index

Newton Polygon, 34 transform normalization, 3, 10, 12, 26 strict, 20, 25, 38, 65 total, 21, 25, 37 ord, 22, 138 weak, 65 order, 3, 19, 25, 138 Tschirnhausen transformation, 24, 28, 51, νJ (q), 63 56, 61, 67, 70 νR, 175 uniformizing parameters, 170 νX , 63, 176 νq(J), 175 upper semi-continuous, 176 νq(X), 175 upper semi-continuous, 176 valuation, 12, 133 dimension, 134 discrete, 134 Piltant, 13 rank, 134 principalization of ideals, 40, 43, 99 rational rank, 136 Puiseux, 6 ring, 134 Puiseux series, 6, 11, 137 variety, 2 integral, 2 resolution of differential forms, 13 quasi-complete, 15 resolution of indeterminacy, 41, 100 subvariety, 2 resolution of singularities, 9, 40, 41, 61, Villamayor, 61 100–102 curve, 10, 12, 26, 27, 30, 38, 39, 42 Walker, 55 embedded, 41, 56, 99 Weierstrass preparation theorem, 24 equivariant, 103 Wiegand, 13 excellent surface, 130 Wlodarczyk, 9, 61 positive characteristic, 10, 12, 13, 30, 42, 105, 152 Zariski, 1, 12, 55, 133, 147, 148, 152, 156 surface, 12, 45, 105, 152 three-fold, 12, 152 resolution of vector fields, 13 ring affine, 2 regular, 7 Rotthaus, 9, 13 scheme Cohen-Macaulay, 11 excellent, 9 non-singular, 7 quasi-excellent, 9 smooth, 7 Schilling, 137 Seidenberg, 13 Serre, 10 Shannon, 151 singularity Sing(J, b), 66 Singb(X), 45, 105, 177 ˆ Singb(OX,q), 177 SNCs, 29, 40, 49 Spivakovsky, 13 surface, 2 tangent space, 8 Teissier, 13 three-fold, 2