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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 mor- phisms. Chapter 8 gives a classification of valuations in algebraic function fields of surfaces, and a modernization of Zariski’s original proof of local uni- formization 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 ba- sic 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 inde- pendent, 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, T`aiH`a,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 stud- ied 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, sur- faces and 3-folds (with char(K) > 5). The first proof in positive characteris- tic 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 .
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