Lectures on Cremona Transformations, Ann Arbor-Rome, 2010/2011

Lectures on Cremona transformations, Ann Arbor-Rome, 2010/2011

Igor Dolgachev

July 4, 2016 ii Contents

1 Basic properties1 1.1 Generalities about rational maps and linear systems...... 1 1.2 Resolution of a rational map...... 3 1.3 The base ideal of a Cremona transformation...... 5 1.4 The graph of a Cremona transformation...... 10 1.5 F -locus and P -locus...... 13

2 Intersection Theory 19 2.1 The Segre class...... 19 2.2 Self-intersection of exceptional divisors...... 25 2.3 Computation of the multi-degree...... 31 2.4 Homaloidal linear systems in the plane...... 33 2.5 Smooth homaloidal linear systems...... 35 2.6 Special Cremona transformations...... 40 2.7 Double structures...... 42 2.8 Dilated transformations...... 49

3 First examples 53 3.1 Quadro-quadric transformations...... 53 3.2 Quadro-quartic transformations...... 56 3.3 Quadro-cubic transformations...... 58 3.4 Bilinear Cremona transformations...... 59 3.5 Monomial birational maps...... 71

4 Involutions 73 4.1 De Jonquieres` involutions...... 73 4.2 Planar Cremona involutions...... 78 4.3 De Jonquieres` subgroups...... 80 4.4 Linear systems of isologues...... 81

iii iv CONTENTS

4.5 Arguesian involutions...... 83 3 4.6 Geiser and Bertini involutions in P ...... 91

5 Factorization Problem 95 5.1 Elementary links...... 95 5.2 Noether-Fano-Iskovskikh inequality...... 108 5.3 The untwisting algorithm...... 115 5.4 Noether Theorem...... 116 5.5 Hilda Hudson’s Theorem...... 120 Lecture 1

Basic properties

1.1 Generalities about rational maps and linear systems

Recall that a rational map f : X 99K Y of algebraic varieties over a field K is a regular map defined on a dense open Zariski subset U ⊂ X. The largest such set to which f can be extended as a regular map is denoted by dom(f). Two rational maps are considered to be equivalent if their restrictions to an open dense subset coincide. A rational map is called dominant if f : dom(f) → Y is a dominant regular map, i.e. the image is dense in Y . Algebraic varieties form a category RatK with morphisms taken to be equivalence classes of dominant rational maps. From now on we restrict ourselves with rational maps of irreducible varieties over C. We use fd to denote the restriction of f to dom(f), or to any open subset of dom(f). A dominant map fd : dom(X) → Y defines a homomorphism of the fields of rational functions f ∗ : R(Y ) → R(X). Conversely, any homomorphism ∗ R(Y ) → R(X) arises from a unique dominant rational map X 99K Y . If f makes R(X) a finite extension of R(Y ), then the degree of the extension is the degree of f. A rational map of degree 1 is called a birational map. It is also can be defined as an invertible rational map. We will further assume that X is a smooth projective variety. It follows that the complement of dom(f) is of codimension ≥ 2. A rational map f : X 99K r Y is defined by a linear system. Namely, we embed Y in a projective space P 0 and consider the complete linear system HY = |OY (1) := |H (Y, OY (1))|. Its ∗ divisors are hyperplane sections of Y . The invertible sheaf fd OY (1) on dom(f) can be extended to a unique invertible sheaf L on all of X. Also we can extend the ∗ 0 sections fd (s), s ∈ V , to sections of L on all of X. The obtained homomorphism ∗ 0 0 0 f : V = H (Y, OY (1)) → H (X, L) is injective and its image is a linear subspace V ⊂ H0(X, L). The associated projective space |V | ⊂ |L| is the linear

1 2 LECTURE 1. BASIC PROPERTIES

r system HX defining a morphism fd : dom(f) → Y,→ P . The rational map f is given in the usual way. Evaluating sections of V at a point, we get a map dom(f) → |V ∨|, and by restriction, the map dom(f) → |V 0∨| which factors through Y,→ |V 0∨|. A choice of a basis in V and a basis in V 0 r defines a rational map f : X 99K Y ⊂ P , where r = dim |HY |. For any rational map f : X 99K Y and any closed reduced subvariety Z of −1 −1 Y we denote by f (Z) the closure of fd (Z) in X. It is called the inverse transform of Z under the rational map f. Thus the divisors from HX are inverse r transforms of hyperplane sections Z of Y in the embedding ι : Y,→ P such that Z ∩ f(dom(f)) 6= ∅. If |V 0| ⊂ |L0| is a linear system on Y , then we define its inverse transform f −1(|V 0|) of |V 0| following the procedure from above defining the linear sys- −1 0 tem HX . The members of f (|V |) are the inverse transforms of members of |V 0|. When f is a morphism, the inverse transform is equal to the full transform f ∗(|V |) ⊂ |f ∗L0|. Let L be a line bundle and V ⊂ H0(X, L). Consider the natural evaluation map of sheaves ev : V ⊗ OX → L defined by restricting global sections to stalks of L. It is equivalent to a map

−1 ev : V ⊗ L → OX whose image is a sheaf of ideals in OX . This sheaf of ideals is denoted b(|V |) and is called the base ideal of the linear system |V |. The closed subscheme Bs(HX ) of X deﬁned by this ideal is called the base locus scheme of |V |. We have

Bs(|V |) = ∩D∈HX D = D0 ∩ ... ∩ Dr (scheme-theoretically), where D0,...,Dr are the divisors of sections forming a basis of V . The largest positive divisor F contained in all divisors from |V | (equivalently, in the divisors D0,...,Dr) is called the fixed component of |V |. The linear system without fixed component is sometimes called irreducible. Each irreducible component of its base scheme is of codimension ≥ 2. If F = div(s0) for some s0 ∈ OX (F ), then the multiplication by s0 defines an injective map L(−F ) → L and the linear map H0(X, L(−F )) → H0(X, L) defines an isomorphism from a subspace V 0 of H0(X, L(−F )) onto V . The linear 0 0 0∨ system |V | ⊂ |L(−F )| is irreducible and defines a rational map f : X 99K |V | equal to the composition of f with the transpose isomorphism |V ∨| → |V 0∨|. The linear system is called basepoint-free, or simply free if its base scheme is ∼ empty, i.e. b(|V |) = OX . The proper transform of such a system under a rational 1.2. RESOLUTION OF A RATIONAL MAP 3 map is an irreducible linear system. In particular, the linear system HX defining a rational map X 99K Y as described in above, is always irreducible. The morphism ∨ U = X \ Bs(HX ) → |V | defined by the linear system |V | is the projection

−1 ∼ ∼ ∨ ∨ Proj Sym(V ⊗ (L|U) ) = Proj Sym(V ⊗ OU ) = U × |V | → |V |.

If |V | is an irreducible linear system,

dom(f) = X \ Bs(HX )red = X \ Supp(Bs(HX )).

Let f : X 99K Y be a rational map deﬁned by the inverse transform HX = |V | of a very ample complete linear system HY on Y . After choosing a basis in 0 r H (Y, OY (1)) and a basis (s0, . . . , sr) in V , the map f : dom(f) → Y,→ P is given by the formula x 7→ [s0(x), . . . , sr(x)].

By definition, this is the formula defining the rational map f : X 99K Y . Of r course, different embeddings Y,→ P define different formulas. Here are some simple properties of the base locus scheme.

(i) |V | ⊂ |L ⊗ b(|V |)| := |H0(X, b(|V |) ⊗ L)|.

(ii) Let φ : X0 → X be a regular map, and V 0 = φ∗(V ) ⊂ H0(X0, φ∗L). Then −1 −1 φ (b(|V |)) = b(f (|V |)). Recall that, for any ideal sheaf a ⊂ OX , its −1 ∗ 0 inverse image φ (a) is deﬁned to be the image of φ (a) = a ⊗OX OX in OX0 under the canonical multiplication map.

(iii) If b(|V |) is an invertible ideal (i.e. isomorphic to OX (−F ) for some effective divisor F ), then dom(f) = X and f is deﬁned by the linear system |L(−F )|.

(iv) If dom(f) = X, then b(|V |) is an invertible sheaf and Bs(HX ) = ∅.

1.2 Resolution of a rational map

Deﬁnition 1.2.1. A resolution of a rational map f : X 99K Y of projective varieties is a pair of regular projective morphisms π : X0 → X and σ : X0 → Y such −1 that f = σ ◦ π (in RatK) and π is an isomorphism over dom(f). 4 LECTURE 1. BASIC PROPERTIES

X0 π σ ~ f X / Y We say that a resolution is smooth (normal) if X0 is smooth (normal).

Let Z = V (a) be the closed subscheme deﬁned by a and

∞ M k σ : BlZ X = Proj a → X k=0 be the blow-up of Z (see [Hartshorne]). We will also use the notation Bl(a) for −1 the blow up of V (a). The invertible sheaf σ (a) is isomorphic to OBlZ X (−E), where E is the uniquely deﬁned effective divisor on BlZ X. We call E the exceptional divisor of σ. Any birational morphism u : X0 → X such that u−1(a) is an invertible sheaf of ideals factors through the blow-up of a. This property uniquely determines the blow-up, up to isomorphism. The morphism u is isomorphic to the 0 morphism BlZ0 X → X for some closed subscheme Z ⊂ Z. The exceptional divisor of this morphism contains the pre-image of the exceptional divisor of σ. For any closed subscheme i : Y,→ X, the blow-up of the ideal i−1(a) in Y is isomorphic to a closed subscheme of BlZ X, called the proper transform of Y under the −1 blow-up. Set-theoretically, it is equal to the closure of σ (Y \ Y ∩ Z) in BlZ X. In particular, it is empty if Y ⊂ Z. + + Let ν : BlZ X → X denote the normalization of the blow-up BlZ X and E be the scheme-theoretical inverse image of the exceptional divisor. It is the exceptional divisor of ν. We have

+ ν∗OBlZ X (−E ) = a¯, where a¯ denotes the integral closure of the ideal sheaf a (see [Lazarsfeld], II, 9.6). A local definition of the integral closure of an ideal I in an integral domain A is the n n−1 set of elements x in the fraction field of A such that x +a1x +...+an = 0 for k + P some n > 0 and ak ∈ I (pay attention to the power of I here). If E = riEi, considered as a Weil divisor, then locally elements in a¯ are functions φ such that ∗ ordEi (ν (φ)) ≥ ri for all i. + m We have BlZ X = BlZ X if and only a is integrally closed for m 0. If X is nonsingular, and dim X = 2, then m = 1 suffices.

Proposition 1.2.1. Let π : BlBs(HX )X → X be the blow-up scheme of the base locus scheme of a rational map f : X 99K Y . Then there exists a unique regular 1.3. THE BASE IDEAL OF A CREMONA TRANSFORMATION 5

map σ : BlBs(HX )X → Y such that (π, σ) is a resolution of f. For any resolution 0 0 0 (π , σ ) of f there exists a unique morphism α : X → BlBs(HX )X such that π0 = π ◦ α, σ0 = σ ◦ α. −1 Proof. By properties (ii) and (iii) from above, the linear system π (HX ) = ∗ −1 |π (L) ⊗ π (b)| defines a regular map σ : BlBs(HX ) → Y . It follows from the definition of maps defined by linear systems that f = σ ◦ π−1. For any res- 0 0 −1 0 olution, (π , σ ), the base locus of the pre-image π (HX ) on X is equal to the 0 pre-image of the base scheme of HX . The morphism σ is defined by the linear 0−1 0 system π (HX ) and hence its base sheaf is invertible. This implies that π factors through the blow-up of Bs(HX ). Note that we also obtain that the exceptional divisor of π0 is equal to the pre- image of the exceptional divisor of the blow-up of Bs(HX ). In many applications we will need a smooth resolution of a rational map. The following result follows from Hironaka’s theorem on resolutions of singularities. P Definition 1.2.2. An effective divisor D = aiDi on a smooth variety of dimension n is called a simple normal crossing (SNC) divisor if each irreducible com- P ponent Di is smooth and, at any point x ∈ Supp(D), the reduced divisor Di is defined by local equations φ1 ··· φk = 0, where (φ1, . . . , φk) is subset of a local system of parameters in OX,x. Theorem 1.2.2. Let X be an irreducible algebraic variety (over C as always) and let D be an effective Weil divisor on X. (i) There exists a projective birational morphism ν : X0 → X, where X0 is smooth and µ has divisorial exceptional locus Exc(ν) such that ν∗(D) + Exc(ν) is a SNC divisor. (ii) X0 is obtained from X by a sequence of blow-ups with smooth centers sup- ported in the singular locus Sing(X) and the singular locus Sing(D) of D. In particular, one can assume that ν is an isomorphism over X \(Sing(X)∪ Sing(D)). We will call X0 a log resolution of (X,D) and will apply this to the case when D is the exceptional divisor of the normalization of the blow-up of a closed subscheme Z in X. We will call it a logresolution of Z.

1.3 The base ideal of a Cremona transformation

Theorem 1.3.1. Assume that f : X 99K Y is a birational map of normal projective varieties and f is given by a linear system HX = |V | ⊂ |L| equal to the inverse 6 LECTURE 1. BASIC PROPERTIES

0 transform of a very ample complete linear system HY on Y . Let (π, σ): X → X × Y be a resolution of f and E be the exceptional divisor of π. Then the canonical map V → H0(X0, π∗L(−E)) is an isomorphism.

Proof. Set b = b(HX ). We have natural maps

V → H0(X, L ⊗ b) → H0(X0, π∗L ⊗ π−1(b))

=∼ 0 0 ∗ =∼ 0 0 ∗ =∼ 0 ∗ → H (X , (π L)(−E)) → H (X , σ OY (1)) → H (Y, σ∗σ OY (1))

=∼ 0 =∼ 0 ∼ → H (Y, OY (1) ⊗ σ∗OX0 ) → H (Y, OY (1)) = V. ∼ Here we used the Main Zariski Theorem that asserts that σ∗OX0 = OY because σ is a birational morphism and Y is normal [Hartshorne Cor. 11.4]. The last r isomorphism comes from the assumption of linear normality of Y in P which 0 ∼ 0 r ∼ gives H (Y, OY (1)) = H (P , OPr (1)) = V . By deﬁnition of the linear system deﬁning f, the composition of all these maps is a bijection. Since each map here is injective, we obtain that all the maps are bijective. One of the compositions is our map V → H0(X0, π∗L(−E)), hence it is bijective.

Corollary 1.3.2. Assume, additionally, that the resolution (X, π, σ) is normal. Then the natural maps

0 0 0 ∗ 0 V → H (X, L ⊗ b(HX )) → H (X , π (L)(−E)) → H (Y, L ⊗ b(HX )) are bijective.

n n We apply Theorem 1.3.1 to the case when f : P 99K P is a birational map, a Cremona transformation. In this case L = OPn (d) for some d ≥ 1, called the degree of the Cremona transformation f. We take HY = |OPn (1)|. The linear system HX = |b(HX )(d)| deﬁning a Cremona transformation is called a homaloidal linear system. Classically, members of HX were called homaloids. More generally, a k-homaloid is a proper transform of a k-dimensional linear subspace in the target space. They were classically called Φ-curves, Φ-surfaces, etc.).

Proposition 1.3.3. 1 n H (P , L ⊗ b(HX )) = 0. 1.3. THE BASE IDEAL OF A CREMONA TRANSFORMATION 7

n Proof. Let (π, σ): X → P be the resolution of f deﬁned by the normalization ∗ ∼ ∗ of the blow-up of Bs(HX ). We know that π L(−E)) = σ OPn (1). The exact sequence

1 n ∗ 1 ∗ 0 n 1 ∗ 0 → H (P , σ∗σ OPn (1)) → H (X, σ OPn (1)) → H (P ,R σ∗σ OPn (1)) deﬁned by the Leray spectral sequence, together with the projection formula, can be rewritten in the form

1 n 1 ∗ 0 n 1 0 → H (P , OPn (1)) → H (X, σ OPn (1)) → H (P ,R σ∗OX0 ⊗ OPn (1)). (1.1) Let ν : X0 → X be a resolution of singularities of X. Then, we have the spectral sequence pq p q p+q E2 = R σ∗(R ν∗OX0 ) ⇒ R (π ◦ ν)∗OX0 . It gives the exact sequence

1 1 1 R π∗(ν∗OX0 ) → R (π ◦ ν)∗OX0 ) → π∗(R ν ∗ OX0 ).

0 n Since X is normal, ν∗OX0 = OX . Since the composition π ◦ ν : X → P is a 1 birational morphism of nonsingular varieties, R (π ◦ν)∗OX0 = 0. This shows that

1 R π∗(ν∗OX0 ) = 0.

1 n Together with vanishing of H (P , OPn (1)),(1.1) implies that H1(X, π∗(L)(−E)) = 0.

It remains to use that the canonical map

1 n ∼ 1 n ∗ 1 ∗ H (P , L ⊗ b(HX )) = H (P , π∗(π (L)(−E))) → H (X, π (L)(−E)) is injective (use Cechˇ cohomology, or the Leray spectral sequence).

Using the exact sequence

0 → b(HX ) → OPn → OPn (1)/b(HX ) → 0, and tensoring it by OPn (d), we obtain Corollary 1.3.4 (Postulation Formula).

n + d h0(O (d)) = − n − 1. V (b(HX )) d 8 LECTURE 1. BASIC PROPERTIES

Remark 1.3.1. We do not know whether

1 n H (P , b(HX )(d)) = 0.

However, if Bs(HX ) is locally complete intersection, then Lemma 4.3.16 and Re- mark 4.3.17 in [Lazarsfeld] implies that 1 n ∼ 1 ∗ H (P , b(HX )(d)) = H (X, π OPn (d)(−E)) and the rest of the argument in the previous proof imply the vanishing. Under the assumption of Proposition 1.3.3, the exact sequence

0 → b(HX )(d) → b(HX )(d) → (b(HX )/b(HX ))(d) → 0, implies that h0(O (d)) = h0(O (d)), V (b(HX )) V (b(HX )) 1 n or, equivalently, H (P , b(HX )(d)) = 0, if and only if 0 h (b(HX )/b(HX ))(d)) = 0. For example, it does not hold if the base scheme is 0-dimensional. Let us recall the following deﬁnition from [Lazarsfeld], Deﬁnition 9.6.15:

Deﬁnition 1.3.1. Let a be an non-zero ideal in OX .A reduction of a is an ideal r ⊂ a such that ¯r = a¯. A reduction is called minimal if is not properly contained in any other reduction of a Proposition 9.6.16 from loc.cit. says that r is a reduction of a if and only if

ν−1(r) = ν−1(a), where ν : X+ → X is the normalization of the blow-up of a. Also note that the radical ideals of r and a coincide. It is known, that a minimal reduction r can be locally generated at a closed point x ∈ V (r) by `(r)x + 1 elements, where `(r)x is equal to the dimension of the ﬁbre of Bl(r) → X at x ([Vasconcelos], Theorem 1.77). Let a ⊂ OX be an integrally closed ideal. Then a has a canonical primary decomposition a = q1 ∩ ... ∩ qk, (1.2) where qi are primary integrally closed ideals. If we write the exceptional divisor of 0 P the normalized blow-up X of a in the form E = riEi, where Ei are irreducible divisors, then qi = ν∗OX0 (−riEi). 1.3. THE BASE IDEAL OF A CREMONA TRANSFORMATION 9

Let ∞ M 0 n B(HX ) = H (P , b(HX )(k)) k=0 be the graded homogeneous ideal of the closed subscheme Bs(HX ).

0 n ∼ 0 n ∼ n+1 B(HX )d = H (P , b(HX )(d)) = H (P , OPn (1)) = C . (1.3) Also we have B(HX )k = 0, k < d. (1.4)

Indeed, otherwise |b(HX )(d)| contains |OPn (d − k)| + |b(HX )(k)| and its dimension is strictly larger than n+1 if k < d−1, or it has ﬁxed component if k = d−1. The two formulas (1.3) and (1.4) are classically known as postulation formulas for homaloidal linear system. The base locus Bs(HX ) could be very complicated, e.g. it could be non- reduced or even contain embedded components. We will see this behavior in later examples. However, there are some special properties they share.

Proposition 1.3.5. Let B(HX )red be the homogeneous ideal of Bs(HX )red. Then

q B(HX ) 6= B(HX )red for any q > 1.

Proof. We set for brevity of notation, B(HX ) = B, B(HX )red = Bred. Assume q q that B = Bred for some q ≥ 2. Let G ∈ Bred, then G = A0F0 + ...ArFr, d where F0 ...Fn is a basis of V . Hence deg G ≥ q . Now each Fi is a sum of d the products Gi1 ··· Giq, where Gij ∈ Bred. Hence each Gij has degree p := q d and p is an integer. Indeed each Gij has degree pij ≥ q and, moreover, we have pi1 + ··· + piq = d. Since deg Gij = p, it follows that the multiplication

q µ : Sym (Bred)p → (Bred)pq

n s is surjective. Let s + 1 = dim(Bred)p, then f = u ◦ g, where g : P → P is the s+q s ( )−1 rational map deﬁned by | (Bred)p | and u : P → P q is the q-th Veronese n map. On the other hand, f is birational, hence it follows that f(P ) is a linear s space contained in the Veronese variety u(P ). But this variety does not contain linear spaces unless q = 1, and the statement follows.

Another special property of Bs(HX ) is given by the following.

Proposition 1.3.6. Bs(HX ) is not a complete intersection. 10 LECTURE 1. BASIC PROPERTIES

Proof. If Bs(HX ) is a complete intersection, then Bs(HX ) is equidimensional of codimension c ≤ n and its homogeneous ideal B(HX ) is generated by c forms G1 ...Gc. By (1.3), we must have B(HX )d = n + 1. If Gi has degree di < d then

0 n GiH (P , OPn (d − di)) ⊂ V.

Then, for dimension reasons, d − di = 1. But then V (Gi) is a ﬁxed component of Bs(HX ), a contradiction. Hence it holds di ≥ d, i = 1, . . . , c. Since c ≤ n it follows that dim B(HX )d ≤ c ≤ n, contradicting (1.3).

n Remark 1.3.2. Let X ⊂ P be a smooth irreducible non-degenerate subvariety of n P . Recall that Hartshorne’s conjecture says that X is a complete intersection as 2n soon as dim X > 3 . So, assuming that this conjecture is true, we obtain

• If the base locus scheme Bs(HX ) is smooth and irreducible, then 2n dim Bs(H ) ≤ . X 3

In many examples Bs(HX )red is smooth but Bs(HX ) 6= Bs(HX )red. An additional quite strong condition is that Bs(HX )red is smooth, integral and moreover, f admits a resolution (π, σ), where π is the blow-up of Bs(HX )red. Such a situation has been studied by Crauder and Katz in [CrauderKatz,Amer. J. Math. (1991)], in particular they show that in this case, assuming Hartshorne’s conjecture,

Bs(HX ) = Bs(HX )red.

They also show that, if Hartshorne’s conjecture holds and n ≥ 7, then d ≤ 4.

1.4 The graph of a Cremona transformation

We define the graph Γf of a rational map f : X 99K Y as the closure in X × Y of the graph Γfd of fd : dom(f) → Y . Clearly, the graph, together with its projections to X and Y , defines a resolution of the rational map f. Note that the switching the orders of the projections, we obtain that the same variety is the graph of the inverse transformation f −1. In another way, we do not switch the projections but replace Γf with the isomorphic variety obtained by switching the factors of the product n n P × P . By the universal property of the graph, we obtain that, for any resolution (X0, π, σ) f X0 → X × Y Γ of , the image of the map contains f|dom(f) . Since X0 is an irreducible projective variety, the image is closed and irreducible, hence it coincides with Γf . Thus the first projection Γf → X has the universal property 1.4. THE GRAPH OF A CREMONA TRANSFORMATION 11 for morphisms which invert b(|V |), hence it is isomorphic to the blow-up scheme Blb((|V |)X. n n Let us consider the case of a Cremona transformation f : P 99K P . If (F0,...,Fn) is a basis of V defining the homaloidal linear system, then the graph n n is a closed subscheme of P ×P which is an irreducible component of the closure n of the subvariety of dom(f) × P defined by 2 × 2-minors of the matrix

F (x) F (x) ...F (x) 0 1 n , (1.5) y0 y1 . . . yn where x = (x0, . . . , xn) are projective coordinates in the ﬁrst factor, and y = (y0, . . . , yn) are projective coordinates in the second factor In the usual way, the graph Γf deﬁnes the linear maps of cohomology

∗ 2k n 2k n fk : H (P , Z) → H (P , Z)

2k n ∼ Since H (P , Z) = Z, these maps are deﬁned by some numbers dk, the vector (d0, . . . , dn) is called the multi-degree of f. In more details, we write the coho- ∗ n n mology class [Γf ] in H (P × P , Z) as

n X i n−i [Γf ] = dih1h2 , k=0

∗ n where hi = pri h and h is the class of a hyperplane in P . Then

∗ k ∗ k k k fk (h ) = (pr1)∗([Γf ] · (pr2(h )) = (pr1)∗(dkh1) = dkh .

The multi-degree vector has a simple interpretation. The number dk is equal to the degree of the proper transform under f of a general linear subspace of codimension n k in P . Since f is birational, d0 = dn = 1. Also d1 = d is the degree of f. Inverting f, we obtain that ˜ Γf −1 = Γf , ˜ n n where Γf is the image of Γf under the self-map of P ×P that switches the factors. −1 In particular, we see that (dr, dr−1, . . . , d0) is the multi-degree of f . In the case when f is a birational map, we have d0 = dn = 1. We shorten the deﬁnition by saying that the multii-degree of a Cremona transformation is equal to (d1, . . . , dn−1).

Remark 1.4.1. Let T be an automorphism of K(z1, . . . , zn) identical on K. This is an algebraic deﬁnition of a Cremona transformation. The automorphism T can be given by n rational functions R1(z1, . . . , zn),...,Rn(z1, . . . , zn). Write each 12 LECTURE 1. BASIC PROPERTIES

0 Ri as the ratio of two coprime polynomials Ri = Ai/Bi. Let di = deg Ai, di = deg Bi. Making a substitution zi = ti/t0, we can write

0 ¯ di−di Ai(t0, . . . , tn) Ri = t0 , i = 1, . . . , n, B¯i(t0, . . . , tn) where A¯i(t0, . . . , tn) and B¯i(t0, . . . , tn) are homogeneous polynomials of degree 0 di and di. After reducing to the common denominator, we see that the correspond- n n ing Cremona transformation f : P 99K P can be given by the homogeneous polynomials

D−d +d0 0 D ¯ ¯ ¯ 1 1 ¯ ¯ ¯ ¯ D−dn+dn ¯ ¯ ¯ (t0 A0B1 ··· Bn, t0 A1B0B2 ··· Bn, . . . , t0 AnB1 ··· Bn−1), where D = d1 + ... + dn. Of course, we have to divide these polynomials by their greatest common divisor. The degree of the transformation is less than or equal 0 0 D + d1 + ... + dn . The next result due to L. Cremona puts some restrictions on the multi-degree of a Cremona transformation. Proposition 1.4.1. For any n ≥ i, j ≥ 0,

1 ≤ di+j ≤ didj, dn−i−j ≤ dn−idn−j.

Proof. It is enough to prove the ﬁrst inequality. The second one follows from the ﬁrst one by considering the inverse transformation. Write a general linear subspace Li+j of codimension i + j as the intersection of a general linear subspace Li of codimension i and a general linear subspace Lj of codimension j. Then, we have −1 −1 −1 f (Li+j) is an irreducible component of the intersection f (Li) ∩ f (Lj). By Bezout’s Theorem,

−1 −1 −1 di+j = deg f (Li+j) ≤ deg f (Li) deg f (Lj) = didj.

There are more conditions on the multi-degree which follow from the irre- ducibility of Γf . For example, we have the following Hodge type inequality (see [Lazarsfeld, Cor. 1.6.3):

Theorem 1.4.2. Let X be an irreducible variety of dimension n, and α1, . . . , αp, β1, . . . , βn−p be the classes of nef Cartier divisors. Then, for any integers 0 ≤ p ≤ n,

p p p (α1 · ... · αp · β1 · ... · βn−p) ≥ (α1 · β1 · ... · βn−p) · ... · (αp · β1 · ... · βn−p). 1.5. F -LOCUS AND P -LOCUS 13

Here we use the intersection theory of Cartier divisors (more about this in the next lecture). For example, letting p = 2, and α1 = α, α2 = β, β1 = ... = βi−1 = α and βi = ... = βn−p = β, we obtain the inequality

2 si ≥ si−1 · si+1,

i n−i where si = α · β . If we take X = Γf and take α = h1, β = h2 restricted to Γf , we get the inequality 2 di ≥ di−1di+1 (1.6) for the multi-degree of a Cremona transformation f. For example, if n = 3, the only non-trivial inequality followed from the Cremona inequlities is d0d2 = d2 ≤ 2 d1, and this is the same as the Hodge type inequality. However, if n = 4, we get new inequalities besides the Cremona ones. For example, (1, 2, 3, 5, 1) satisﬁes the Cremona inequalities, but does not satisfy the Hodge type inequality. The following are natural questions related to the classiﬁcation of possible multi-degrees of Cremona transformations.

• Let (1, d1, . . . , dn−1, 1) be a sequence of integers satisfying the Cremona inequalities and the Hodge type inequalities: Does there exist an irreducible n n P k n−k reduced close subvariety Γ of P × P with [Γ] = dkh1h2 ?

• What are the components of the Hilbert scheme of this class containing an integral scheme?

n n Note that any irreducible reduced closed subvariety of P × P with multi-degree (1, d1, . . . , dn−1, 1) is realized as the graph of a Cremona transformation.

Remark 1.4.2. The inequalities (1.6) show that the sequence (d0, d1, . . . , dn) is a log-concave sequence. Many other situations where such sequences arise are discussed in June Huh’s preprint on the archive.

1.5 F -locus and P -locus

n Let (X, π, σ) be any normal resolution of a Cremona transformation f : P − → n P . It factors through the blow-up B(f) of the integral closure of the ideal sheaf of −1 the base scheme of b(HX ) as well as the blow-up B(f ) of the integral closure of the ideal sheaf of the base scheme of the homaloidal linear system b(|W |) deﬁning the inverse Cremona transformation f −1. So we have a commutative diagram 14 LECTURE 1. BASIC PROPERTIES

| # B(f) / B(f −1)

! { 0 ν Γf ν p1 p2

nÙ | f # n P / P

P n Let E = i∈I riEi be the exceptional divisor of ν : B(f) → P and F = P 0 −1 n 0 j∈J mjFj be the exceptional divisor of ν : B(f ) → P . Let J be the largest 0 subset of J such that the proper transform of Fj, j ∈ J , in X is not equal to the P proper transform of some Ei in X. The image of the divisor j∈J0 Fj under the −1 p1 n composition map B(f ) → Γf → P is classically known as the P -locus of n f (the F -locus is Bs(HX )red). It is hypersurface in the source P . The image of any irreducible component of the P -locus is blown down under f (after we restrict ourselves to dom(f)) to an irreducible component of the base locus of f −1. If we n+1 n+1 consider f as a regular map f˜ : C → C deﬁned by the same formula. Then n the P -locus is the image in P of the set of critical points of f˜. It is equal to the set of zeroes of the determinant of the Jacobian matrix of f˜ ∂F J = i . ∂tj i,j=0,...,n

So we expect that the P -locus is a hypersurface of degree (d − 1)n+1. Some of its components enter with multiplicities. We assign these multiplicities to the corresponding irreducible components of the P -locus. Example 1.5.1. Consider the standard quadratic transformation given by

Tst :[x0, x1, x2] 7→ [x1x2, x0x2, x0x1]. (1.7)

The P -locus is the union of three coordinate lines V (ti). The Jacobian matrix is   0 t2 t1 J = t2 0 t0 . t1 t0 0

Its determinant is equal to 2t0t1t2. We may take X = B(f) as a smooth resolution of f. Let E1,E2,E3 be the exceptional divisors over the base points 1.5. F -LOCUS AND P -LOCUS 15 p1 = [1, 0, 0], p2 = [0, 1, 0], p3 = [0, 0, 1], and Li, i = 1, 2, 3, be the proper transform of the coordinate lines V (t0),V (t1),V (t2), respectively. Then the mor- 2 phism σ : X → P blows down L1,L2,L3 to the points p1, p2, p3, respectively. Note that f −1 = f, so there is no surprise here. Recall that the blow-up of a closed subscheme is deﬁned uniquely only up to an isomorphism. If we identify B(f) −1 −1 with B(f ), the morphism B(f) 99K B(f ) in the above diagram is an auto- morophism induced by the Cremona transformation. The surface B(f) is a Del Pezzo surface of degree 6, a toric Fano variety of dimension 2. The complement of the open torus orbits is the hexagon of lines E1,E2,E3,L1,L2,L3 intersecting each other as in the following picture. We call them lines because they become 6 lines in the embedding X,→ P given by the anti-canonical linear system. The automorphism of the surface is the extension of the inversion automorphism z → z−1 of the open torus orbit to the whole surface. It deﬁnes the symmetry of the hexagon which exchanges its opposite sides.

L3 E2

E1 L1

L2 E3

Now let us consider the ﬁrst degenerate standard quadratic transformation given by 0 2 Tst :[x0, x1, x2] 7→ [x1, x0x1, x0x2]. (1.8)

The P -locus consists of the line V (t0) blown down to the point p1 = [1, 0, 0] and the line V (t1) blown down to the point p2 = [0, 0, 1]. The Jacobian matrix is   0 2t1 0 J = t1 t0 0  . t2 0 t0

2 Its determinant is equal to −2t0t1. Thus the line V (t1) enters with multiplicity 2. Let us see what is the resolution in this case. The base scheme is smooth 2 at p1 and locally isomorphic to V (y , x) at the point p2, where y = t1/t2, x = 16 LECTURE 1. BASIC PROPERTIES

0 t0/t2. The blow-up B(f) is singular over p2 with the singular point 21 of type A1 corresponding to the tangent direction t0 = 0. Thus the exceptional divisor of 2 B(f) → P is the sum of two irreducible components E1 and E2 both isomorphic 1 0 to P with the singular point p2 lying on E1. The exceptional divisor of B(f) = −1 2 B(f → P is the union of two components, the proper transform L1 of the line 0 V (t1) and the proper transform L2 of the line V (t0). When we blow-up p2, we get 2 a smooth resolution X of f. The exceptional divisor of π : X → P is the union 0 of the proper transforms of E1 and E2 on X and the exceptional divisor E1 of the 2 blow-up X → B(f). The exceptional divisor of σ : X → P is the union of the 0 proper transforms of L1 and L2 on X and the exceptional divisor E1. Note that 0 the proper transforms of E1,E2 and L1,L2 are (−1)-curves and the curve E1 is a (−2)-curve.1

L2 −1 E0 1 −2

E1 −1 −1 E2 L1 −1

Finally, we can consider the second degenerate standard quadratic transformation. Its reduced base scheme consists of one point. The map is given by the formula

00 2 2 Tst :[x0, x1, x2] 7→ [x0, x0x1, x1 − x0x2]. (1.9)

Its unique base point is [0, 0, 1]. In afﬁne coordinates x = x0/x2, y = x1/x2, the base ideal is (x2, xy, y2−x) = (y3, x−y2). The blow-up of this ideal is singular. It ∼ 1 has a singular point of type A2 on the irreducible exceptional divisor E = P . The P -locus consists of one line x0 = 0. A smooth resolution of the transformation is obtained by blowing up twice the singular point. The exceptional divisor of this blow-up is the chain of two (−2)-curves. So the picture is as follows:

1a (−n)-curve is a smooth rational curve with self-intersection −n. 1.5. F -LOCUS AND P -LOCUS 17

L −1 E00 −2

E0 −2 E −1

00 Note that the jacobian matrix of transformation Tst is equal to   2t0 0 0  t1 t0 0  −t2 2t1 −t0

3 Its determinant is equal to −2t0. So the P-locus consists of one line V (t0) taken with multiplicity 3. 18 LECTURE 1. BASIC PROPERTIES Lecture 2

Intersection Theory

2.1 The Segre class

The multi-degree of the graph Γf can also be computed using intersection theory. Recall some relevant theory from [Fulton]. Let X be a scheme of finite type over a field K.A k-cycle is an element of the free abelian group Zk(X) generated by the set of points x ∈ X of dimension k (i.e. the residue field κ(x) of x is of algebraic dimension k over K). We identify a point x with its closure in X, it is an irreducible reduced closed subscheme V of dimension k (a subvariety). We write [V ] for V considered as an element of 0 Zk(X). Two k-cycles Z and Z are called rationally equivalent if the difference is 1 equal to the projections of a cycle Z(0) − Z(∞) on X × P for some cycle Z on the product. One can give an equivalent definition as follows. A prime divisor p of height 1 in an integral domain A defines a function ordp : A\{0} → Z by setting ordp(a) = length(Ap/(ap), where ap is the image of a in the local (one-dimensional) ring Ap. ∗ This function is extended to a unique homomorphism Q(A) → Z, where Q(A) is the field of fractions of A. By globalizing, we obtain a function ordx : R(X) → Z, where x is a point on X of codimension 1 and R(X) is the field of rational func- 1 tions on a variety X . Now we define the subgroup Bk(X) of rationally equivalent P k-cycles as the group generated by cycles of the form x ordxφx, where φ is a rational function on a subvariety Y ⊂ X of dimension k + 1. The quotient group Ak(X) = Zk(X)/Bk(X) is called the Chow group of k-cycles on X. We set

A∗(X) = ⊕kAk(X).

1 From now on a variety over a ﬁeld K means an integral algebraic scheme over K

19 20 LECTURE 2. INTERSECTION THEORY

P For any α = nxx ∈ Z0(X), we deﬁne

Z X α = nx[κ(x): K]. X x

When X is proper, this extends to A0(X), and, to the whole A∗(X), where, by R definition, X α = 0 if α ∈ Ak(X), k > 0. For any scheme X one defines its fundamental class by X [X] = nV [V ], where V is an irreducible component of X and nV is its multiplicity, the length of OX,η, where η is a generic point of V . For any proper morphism σ : X → Y of schemes, one defines the push- forward homomorphism σ∗ : A∗(X) → A∗(Y ) by setting, for any subvariety V ,

σ∗[V ] = deg(V/σ(V ))[σ(V )] and extending the definition by linearity. Note that deg(V/σ(V )) = 0 if the map V → σ(V ) is not of finite degree. One checks that rationally equivalent to zero cycles go to zero, so the definition is legal. The homomorphism σ∗ preserves the grading of A∗(X). ∗ The pull-back σ : A∗(Y ) → A∗(X) is defined only for flat morphisms, regular closed embeddings, and their compositions. For a flat morphism σ one sets, for any subvariety V , σ∗[V ] = [σ−1(V )]. It shifts the degree by increasing it by the relative dimension of σ. Recall that a Weil divisor on a variety X of dimension n is an element of ∗ Zn−1(X).A Cartier divisor on X is a section of the sheaf RX /OX , where RX is the constant sheaf of total rings of fractions. The function ordx : R(X) → Z ∗ factors through OX and defines a homomorphism X CDiv(X) → WDiv(X),D 7→ [D] = ordx(D)x, x where CDiv(X) (resp. WDiv(X)) is the group of Cartier (resp. Weil) divisors on X. Let D be a Cartier divisior on X. One can restrict it to any subvariety V of X, and set D · [V ] = [j∗(D)], 2.1. THE SEGRE CLASS 21 where j : V,→ X is the inclusion morphism of V to X. It is considered as a cycle on V and also as a cycle on X by means of j∗ : A∗(V ) → A∗(X). This extends by linearity to the intersection D · α of D with any cycle class α ∈ A∗(X), so we can consider any divisor D as an endomorphism α 7→ D · α of A∗(X). It depends only on the linear equivalence class of D. By iterating the endomorphism, we can define, for any Cartier divisors D1,...,Dk and α ∈ Am(X), the intersection

D1 · ... · Dk · α ∈ Am−k(X). For any closed subvariety Y containing Supp(D) ∩ V we can identify D · [V ] with an element of A∗(Y ). In particular, for any Y as above, any Cartier divisor D on X can be considered as a homomorphism

Ak(Y ) → Ak−1(Supp(D) ∩ Y ), α 7→ D · α. By deﬁnition, D1 ··· Dk = D1 ····· Dk · [X]. k If X is of pure dimension n, this is an element of An−k(X). We abbreviate D = D ··· D (k times). The intersection of Cartier divisors is commutative, associative, the projection formula holds, and depends only on the linear equivalence classes. Let j : Y,→ X be a closed subscheme of a scheme Y deﬁned by an ideal sheaf IY . Let ∞ M k k+1 C = CY X = Spec IY /IY k=0 be the normal cone of Y in X. Let ∞ M k k+1 P (C ⊕ OY ) = Proj ( IY /IY ⊗ OY [z] , k=0 where the tensor product is the tensor product of graded OY -algebras with deg z = 1. This is the projectivized normal cone of Y in X. Recall the standard exact sequence of sheaves of differentials:

d I /I2 → j∗(Ω1 ) → Ω1 → 0. (2.1) Y Y X/K Y/K Assume that Y = x is a closed point of X. Then, Ω1 = 0. By localizing at x, Y/K we obtain an isomorphism ([Hartshorne], Chapter 2, Prop. 8.7)

m /m2 ∼ Ω1 (x). X,x X,x = X/K

The dual of the first space is, by definition, the Zariski tangent space TX,x of X at x. Going back to the general case, and passing to the fibres of the sheaves, for 22 LECTURE 2. INTERSECTION THEORY any closed point x ∈ X, we get an injection TY,x → TX,x with cokernel equal to 2 ∨ a subspace of (IY (x)/IY (x) ) . In many cases, the kernel of the map d is trivial (e.g. when X is reduced and a locally complete intersection in characteristic 0, 2 ∨ or when X,Y are nonsingular), hence (IY (x)/IY (x) ) can be interpreted as the fibre of the normal bundle of Y in X at the point x. More precisely, by definition, the X-scheme (Ω1 ) := Spec Sym(Ω1 ) V X/K X/K is the tangent vector bundle of X (its fibres at closed points are Zariski tangent spaces). The Y -scheme Spec Sym(j∗(Ω1 ) is the restriction (Ω1 )| of the X/K V X/K Y tangent bundle (Ω1 ) of X to Y . The surjection of sheaves j∗(Ω1 ) → Ω1 V X/K X/K Y/K defines a surjection of the symmetric algebras, and the closed embedding (Ω1 ) ,→ (Ω1 )| . V Y/K V X/K Y In the case when both X and Y are nonsingular, the sheaves of differentials are locally free of ranks equal to the dimensions and the normal cone CY X is also 2 2 a vector bundle isomorphic to V(IY /IY ); = Spec (Sym(IY /IY ). In a general situation, none of these relative affine schemes is a vector bundle, and the normal cone should be considered as the analogue of the normal bundle in a nonsingular situation. The projectivized normal cone is the analog of the projectivization of the nor- ∞ k k+1 mal vector bundle. Let Gra = ⊕k=0IY /IY . The closed subscheme defined by z = 0 in P (C ⊕ OY ) is identified with the exceptional divisor of the blow-up + scheme BlY X. The complement D (z) is equal to CY X. Thus P (C ⊕ OX ) is the analog of the associated projective bundle, and the exceptional divisor is the analog of the “section at infinity”. The canonical projection q : CY X → X admits a section disjoint from the hyperplane at infinity. It corresponds to the surjection k k+1 GrX/Y → OX with kernel ⊕k≥1IY /IY . Recall that, for any graded sheaf of algebras A = ⊕k=0Ak on a scheme Z, ∞ the A-module A[1] = ⊕k=0Ak+1 defines a coherent sheaf O(1) on the projective spectrum Proj A. It is an invertible sheaf if A is generated by A1. In our case, the k algebra GrY/X ⊗ OY [z], and the Rees algebra ⊕k≥0IY satisfy this property, so the projective spectrums come equipped with the corresponding invertible sheaf O(1). ∼ We have O(1) = O(D) for some divisor D. In the case of P(C ⊕ OY ), the divisor D can be taken to be the hyperplane at infinity H∞. In the case of the blow-up scheme BlY X, we can take D to be −E, where E is the exceptional divisor of the blow-up. The Segre class of Y in X, denoted by s(Y,X), is defined to be

X i s(Y,X) = q∗( c1(O(1)) ). i≥0 2.1. THE SEGRE CLASS 23

In the case when j : Y,→ X is a regular embedding, the Segre classes are ex- ∨ pressed in terms of the Chern classes of the conormal sheaf NY/X . Recall that, for any locally free sheaf E of rank r over an n-dimensional variety X, the Chern classes ci(E) are elements of An−i(X) defined as follows. Let P = Proj (Sym E ⊗ OX [z]) be be projectivization of the vector bundle V(E) = Spec Sym E. Let ∗ π : P → X be the natural projection. It is a flat morphism, so π : Ak(X) → ∗ Ak+r(P ) is well-defined. One can show that A∗(P ) is generated by π (A∗(X)) and h = c1(OP (1)) with one basic relation of the form

r r−1 ∗ (−h) + a1(−h) + ··· + ar = 0, ai ∈ π (An−i(X)).

∗ By definition, the i-th Chern class ci(E) is defined by ai = π (ci(E)). Note that here the products are defined in the sense of intersection of Cartier divisor classes i h with any element of A∗(P ). We set r X c(E) = ci(E) ∈ A∗(X). i=0 The following are the basic properties of the Chen classes (see [Hartshorne, Ap- pendix].

(i) c1(OX (D)) = [D];

∗ ∗ (ii) for any morphism σ : Y → X, ci(σ (E)) = σ (c(E)); We deﬁne

∗ ci(E) · [V ] = ci(j E)).

This extends to the intersection ci(E) · α with any α ∈ A∗(X). In particular, one can deﬁne the value of any integral polynomial at the Chern classes of vector bundles and the intersection of this polynomial with any α ∈ A∗(X).

(iii) if 0 → E1 → E2 → E3 → 0 is an exact sequence of locally free sheaves, then c(E2) = c(E1) · c(E3);

∨ i (iv) ci(E ) = (−1) ci(E); Vr (v) c1(E) = c1( E).

We deﬁne the Segre classes si(E) of E by computing them inductively from the relation dim X X c(E) · ( si(E)) = [X]. i=0 24 LECTURE 2. INTERSECTION THEORY

More explicitly, s0(E) = [X], and the rest are found by solving inductively the relations X ci(E)sj(E) = 0, k = 1,..., dim X. i+j=k

Now we can state some properties of s(Y,X).

2 L∞ k k+1 ∼ ∨ • If NY/X = IY /IY is locally free and k=0 IY /IY = Sym NY/X (in this case we say that Y is regularly embedded), then

−1 s(Y,X) = c(NY/X ) . (2.2)

• If f : X0 → X is a morphism of pure-dimensional schemes and Y 0 ⊂ X0 is the pre-image of a closed subvariety Y of X with the induced morphism g : Y 0 → Y , then

0 0 0 g∗(s(Y ,X )) = deg(X /X)s(Y,X). (2.3)

• If f : X0 → X is ﬂat, and Y 0 = f −1(Y ), then

g∗(s(Y,X)) = s(Y 0,X0). (2.4)

• If π : X˜ = BlY X → X is the blow-up of a proper closed subscheme Y in X, and E is the exceptional divisor with the projection σ : E → Y , then

X i−1 i X i−1 i s(Y,X) = (−1) σ∗(E ) = (−1) π∗(E ). (2.5) i≥1 i≥1

i Recall that, by deﬁnition, E ∈ Adim X−i(X˜) which we can also assume to be in i Adim X−i(E). Thus we may also consider π∗(E ) to be in either in A∗(Y ) or equal i to its image π∗(E ) in A∗(X)k under the map i∗ : A∗(Y ) → A∗(X) corresponding to the inclusion morphism j : Y,→ X. Note that the last formula gives

i i−1 σ∗(E ) = (−1) s(Y,X)dim X−i, (2.6) where s(Y,X)k is the component of s(Y,X) in A∗(Y )k, or its image in A∗(X)k. 2.2. SELF-INTERSECTION OF EXCEPTIONAL DIVISORS 25

2.2 Self-intersection of exceptional divisors

The properties of Segre classes allow us to compute the self-intersection of exceptional divisors of blow-ups. Recall that, if X is a nonsingular surface, and Y a closed point on it, the 2 exceptional curve of σ : BlY X → X is a (−1)-curve , i.e. a smooth rational curve E with self-intersection equal to −1. Of course, this agrees with formula 2 −1 s(Y,X)0 = −σ∗(E ) because s(Y,X) = c(NY/X ) = [Y ]. If we replace X by any smooth n-dimensional variety, and take Y to be a closed point, we obtain, similarly, En = (−1)n−1. (2.7) Of course, this can be computed without using Segre classes. We embed X in a projective space, take a smooth hyperplane section H passing through the point Y . Its full transform on BlY X is equal to the union of E and the proper transform H0 n−1 intersecting E along a hyperplane L inside E identiﬁed with P . Replacing H with another hyperplane H0 not passing through Y , we obtain

0 2 H · E = (H0 + E) · E = e + E = 0, where e is the class of a hyperplane in E. Thus E2 = −e. This of course agrees with the general theory. The line bundle O(1) on the blow-up is isomorphic to O(−E). The conormal sheaf OE(−E) is isomorphic OE(1) on the exceptional divisor. 2 2 We also have H0 · E = −H0 · e = −e , hence

2 0 2 2 3 0 = E · H = E · (H0 + E) = E · H0 + E gives E3 = e3. Continuing in this way, we ﬁnd

Ek = (−1)n−1ek. (2.8)

Now let us take Y to be any smooth subvariety with the normal sheaf NY/X . Applying (2.6), we ﬁnd that Z n n−1 n−1 E = (−1) s(Y,X)0 = (−1) s(Y,X). (2.9) X n For example, let X = P and Y be a linear subspace of codimension k > 1. We have ∼ k NY/X = OY (1) ,

2Also known as an exceptional curve of the ﬁrst kind. 26 LECTURE 2. INTERSECTION THEORY hence

n−1 1 1 1 (k−1) X s(Y,X) = = = m (−h)m−k+1. (1 + h)k (k − 1)! 1 − x | k−1 x=−h m=k−1

(note that hi = 0, i > dim Y = n − k). In particular, we obtain

n k−1n−1 E = (−1) k−1 . (2.10)

For example, the self-intersection of the exceptional divisor of the blow-up of a n n line in P is equal to (−1) (n − 1). One can use the exact sequence of sheaves of differentials (2.1) to compute NY/X , where j : Y,→ X is a regular embedding of smooth varieties. It gives

1 ∨ 1 ∨ −1 c(NY/X ) = c((ΩX ) )c((ΩY ) ) .

n For example, when Y = C is a curve of genus g and degree deg C in X = P , we obtain

c1(NC/Pn ) = −KPn · C + KC = (n + 1) deg C + 2g − 2. (2.11)

Another useful formula to compute the Chern classes of a regularly embedded subvariety is the following exact sequence of sheaves on a closed subvariety i : Z,→ Y . ∗ 0 → NZ/Y → NZ/X → i NY/X → 0. (2.12) Passing to the Chern classes and taking the inverse, we get

−1 ∗ −1 s(Z,X) = c(NZ/Y ) · j c(NY/X ) .

We will use this formula many times in the following computations. Recall also from the theory of surfaces that the self-intersection of the proper transform of a curve under the blow-up of a closed point x decreases by the square of the multiplicity of the curve at x. This is generalized as follows (see [Fulton], Appendix B). First let recall the deﬁnition of the proper transform of a closed subscheme Y ⊂ X under the blow-up of the closed subscheme Z of X (see [Hartshorne], Chap. III, Corollary 7.15).

Lemma 2.2.1. Let σ : X0 = Bl(a) be the blow-up of an Ideal a in a noetherian scheme X, and f : Y → X be a morphism of noetherian schemes. Let ν : Y 0 → Y 2.2. SELF-INTERSECTION OF EXCEPTIONAL DIVISORS 27 be the blow-up of the inverse image f −1(a). Then there exists a unique morphism f˜ : Y 0 → X0 such that the following diagram is commutation

f˜ Y 0 / X0

ν σ f Y / X Moreover, if f is a closed embedding, so is f˜. When f : Y,→ X is a closed embedding, the image of f˜ is called the proper transform (or strict transform) of f(Y ). Note that, when f −1(a) = 0 (i.e. j factors through V (a)), the proper transform is the empty scheme. On the other hand, if Y is reduced, and U = Y \ V (f −1(a)) is dense in Y , then Y 0 is isomorphic to the closure of f −1(U) in X0. Note that, in general, the commutative diagram is not a Cartesian diagram, i.e. 0 0 the natural morphism Y → X ×X Y is not an isomorphism. When f is a closed 0 0 −1 embedding, the image of X ×X Y in X is equal, by deﬁnition, to σ (Y ). Its ideal sheaf is equal to the product JE ·JY 0 , where E is the exceptional divisor of σ. Also, it follows from the commutativity of the diagram that f˜−1(σ−1(a)) = ν−1(f −1(a)). This means that the pre-image of the exceptional divisor E of σ under f˜ is equal to the exceptional divisor of ν. For example, take X to be a nonsingular surface, and a to be the ideal sheaf of a closed point x. Let f : Y,→ X be the inclusion morphism of a reduced curve Y passing through x with multiplicity m. Then f −1(a) is the ideal of x considered as a closed point of Y . The blow-up of this ideal in Y has the exceptional divisor ∼ 1 0 equal to a subscheme of the exceptional divisor E = P of X → X of length m. This is the intersection scheme of the proper transform of C with E. Proposition 2.2.2. Let Z,→ Y and Y,→ X be regular embeddings. Let σ : X0 = BlZ X → X be the blow-up of Z and Y be the proper transform of Y under σ. Then ∼ ∗ ∗ NY /X0 = ν (NY/X ) ⊗ j OX0 (−E), where j : Y,→ X0, E is the exceptional divisor of σ and ν : Y → Y is the restriction of σ to Y . In terms of the intersection theory, the formula reads 2 Y = σ∗(Y 2) − Y · E. (2.13) This gives 3 2 Y = Y · σ∗(Y 2) − Y · E = Y 3 − (σ∗(Y 2) − Y · E) · E = Y 3 + Y · E2, 28 LECTURE 2. INTERSECTION THEORY

∗ 2 2 3 where we used the projection formula Y ·σ (Y ) = Y ·σ∗(Y ) = Y . Continuing in this way, we ﬁnd, for any k = 0, . . . , n,

k k−1 Y = σ∗(Y k) + (−1)k−1Y · E = σ∗(Y k) + (−1)k−1Y · Ek. (2.14)

In the case of surfaces, in order the assumptions of Proposition 2.2.2 are satisﬁed, m we have to take Z to be the ideal sheaf Jx of the “fat point” x, where m is the m multiplicity of Y at x. Then the embedding V (Jx ) ,→ Y is regular (check it!). The blow-up of the fat point x is isomorphic to the blow-up BlxX of the point x, but the exceptional divisor is equal to mE, where E is the exceptional divisor of equal to BlxX. We have

2 ∗ deg NY/X = deg OY (Y ) = Y = deg σ (NY/X ).

Also, deg OY (−E) = −E · Y = −multZ Y . This gives

2 2 ∗ 2 2 Y = deg NY /X0 = Y − deg j OX0 (−mE) = Y − m .

3 Next, let us give a less simple example. Let X = P and Y be a connected union of lines `1, . . . , `k with #`i ∩ (∪j6=i`j) = ni. Let p1, . . . , pN be the intersection points and aj be the number of lines containing the point pj. 3 Let σ : X1 → P be the blow-up of the intersection points (in any order) and ν : X → X1 be the blow-up of the proper transforms of the lines (again in any order). Let E be the exceptional divisor of π = ν ◦ σ. The exceptional divisor of σ −1 2 consists of the disjoint sum of N divisors E(pj) = σ (pj) isomorphic to P . Let `¯i be the proper transform of `i in X1. By Proposition 2.2.2,

N ∼ O (1 − n ) ⊕ O (1 − n ). `¯i/X1 = `¯i i `¯i i We have s(`¯ ,X ) = c(N )−1 = (1 + (2 − 2n )[ ])−1. i 1 `¯i/X1 i point Applying (2.10), we obtain

2 Ei = −hi, Z 3 ¯ Ei = s(ì,X1) = 2ni − 2, X1 −1 ¯ ¯ where Ei = ν (ì) and hi = c1(OEi (1)) ∈ A1(Ei). The proper transform E(pj) 2 of E(pj) in X is isomorphic to the blow-up of P at aj points, the intersection point of E(pj) with `¯i. The class in A∗(X) of the corresponding exceptional curve 2.2. SELF-INTERSECTION OF EXCEPTIONAL DIVISORS 29 is equal to the class fj of a fibre of the projection Ei → `¯i. The proper transform ∗ E¯(pj) is equal to the full transform ν (E(pj)). We have

2 3 3 E¯(pj) = −e(pj), E¯(pj) = E(pj) = 1, where e(pj) is the class of a line in E(pj) which we identify with the class in A∗(X). We have ¯ 2 2 ¯ ¯ E(pj) · Ei = −e(pj) · Ei = 0,Ei · E(pj) = −hi · E(pj) = −1.

Here we used that E¯(pj) is isomorphic to the blow-up of aj points. It intersects Ei with pj ∈ ì at one of the exceptional curves which is a fibre of the projection Ei → ì. Now, consider the divisor

k N X X D = αiEi + βjE¯(pj). i=1 j=1 Collecting all together, we get

k N k 3 X 3 3 X 3 ¯ 3 X X 2 2 ¯ D = αj Ei + βj E(pj) + 3 αi βjEi · E(pj) (2.15) i=1 j=1 i=1 j:pj ∈`i k N k X 3 X 3 X X 2 = 2αj (ni − 1) + βj − 3 αi βj. i=1 j=1 i=1 j:pj ∈`i

3 Let H ∈ A∗(X) be the class of the pre-image of a hyperplane in P . Suppose the linear system |rH − D| has no ﬁxed components and deﬁnes a regular map m n f : X 99K P . Then (rH − D) is equal to the product of the degree of the image of X and the degree of the map. So, if (rH − D)3 = 1, we obtain that the map 3 −1 3 3 f is birational onto P . Then the composition f ◦ π : P 99K P is a Cremona transformation. Taking all of this into account, we obtain

k 3 3 3 2 3 3 X (rH − D) = r − D + 3rHD = r − D − 3r αi (2.16) i=1 k k N k 3 X X 3 X 3 X X 2 = r − 3r αi − 2αj (ni − 1) − βj + 3 αi βj. i=1 i=1 j=1 i=1 j:pj ∈`i

Unfortunately, in many cases, the linear system |dH − D| is not base-point free. So, one needs to blow-up more. The situation is analogous to the planar 30 LECTURE 2. INTERSECTION THEORY case, where we have to blow-up inﬁnitely near points (see the next section). For example, suppose σ : X0 → X is the blow-up a closed point x ∈ X and then we want to blow-up a k-codimensional linear subspace L in the exceptional divisor −1 ∼ n−1 E = σ (x) = P . We use formula (2.12) to obtain

∗ c(NL/X0 = c(NL/E) · j (c(NE/X0 )), ∼ where j : L,→ E is the inclusion map. Since NE/X0 = OE(−1) and NL/E = ⊕k OL(1) , we obtain

⊕k k c(NL/X0 ) = c(OL(1) ) · c(OL(−1)) = (1 + h) (1 − h), where h is the class of a hyperplane in L. This gives

n−1−k n−2 0 −k −1 X k X m m−k+1 s(L, X ) = (1 + h) (1 − h) = ( h )( k−1 (−h) ). k=0 m=k−1

This allows us to compute F n, where F is the exceptional divisor of the blow-up of L. For example, let dim X = 3 and L be a line in E. We get s(L, X0) = (1 − 2 −1 3 ∼ h ) = 1. Thus F = 0. Exact sequence (2.12) gives NL/X0 = OL(1)⊕OL(−1) ∼ (it is easy to see that it splits). Thus F is isomorphic to P(OL(1) ⊕ OL(−1)) = ∼ 2 P(OL ⊕ OL(−2)) = F2. We have F = −(f + e), where f is the class of a ﬁbre and e is the class of the exceptional section. Using (2.14), we obtain E¯2 = E2 − E¯ · F = −2[`], where [`] is the class of a line in E¯. Since F · [`] = 1, and F 3 = −F ·(f +s) = 1−F ·s = 0 implies that F ∩E¯ = [s] (considered as a cycle on F ). Thus the two exceptional divisors F and E¯ intersect along the exceptional 3 3 2 section of F2. Now, we get E¯ = E + E¯ · F = 1 + s · F = 1 + 1 = 2. To compute (rH −D)3, we use that H2D = 0 (because a general line does not 2 Pk 2 intersect any line `i). Also, we use that H · D = − i=1 αi (because Ei = −hi and H · hi = 1). Let us consider some special cases. For example, take k = 2,N = 1, α1 = α2 = β1 = 1. We get

(2H − E)3 = 8H3 − E3 + 12H2 · E + 6H · E2 = 8 + 5 − 12 = 1.

The linear system |2H−E| has base locus equal to the line ` in E¯(p) corresponding to the plane spanned by `1 and `2. Using exact sequence (2.12), one checks that the normal bundle of ` is isomorphic to O`(−1) ⊕ O`(−1). The exceptional divisor 1 1 of the blow-up is isomorphic to P × P . One can perform a flop along ` to obtain anew 3-fold X0 on which our linear system has no based points and defines a 3 regular birational morphism to P . To perform the flop, we blow-up `, and then 2.3. COMPUTATION OF THE MULTI-DEGREE 31 blow-down the exceptional divisor along the other ruling. We will discuss flops later. We will see that this leads to a quadro-quadratic transformation which we will consider later. Another example, take ì to be the six edges of the coordinate tetrahedral V (t0t1t2t3). Take 6 4 X X D = Ei + 2 E¯(pj). i=1 j=1

We get D3 = 12 + 32 − 72 = −28. This gives (3H − E)3 = 27 + 28 − 54 = 1. 3 Thus the linear system |3H − E| deﬁnes a birational map f : X 99K P . It gives an example of a cubo-cubic transformation which we will consider later.

3 Definition 2.2.1. A configuration of lines in P is called a homaloidal configuration 3 if there exists numbers (r, αi, βj) such that (dH − D) = 1 and the linear system |dH − D| is basepoint-free and dimension n.

3 Problem 2.2.1. Find all homaloidal conﬁgurations of lines in P .

2.3 Computation of the multi-degree

For all rational smooth varieties X over C we will be dealing with, A∗(X) can be identified with a subgroup of H∗(X, Z) by assigning to any irreducible subvariety ∗ V , its fundamental cycle [V ], or its dual class in H (X, Z) in cohomology if X is ∗ smooth. The operations σ∗ and σ coincide with the corresponding definitions in topology. So, in the definition of the multi-degree of f, we may replace H∗ with A∗. Let us apply the intersection theory to compute the multi-degree of a Cremona transformation. n n Let (X, π, σ) be a resolution of a Cremona transformation f : P 99K P . n n Consider the map ν = (π, σ): X → P × P . We have ν∗[X] = [Γf ], and, by the projection formula,

∗ k n−k k n−k ν (h1h2 ) ∩ [X] = [Γf ] · (h1h2 ) = dk.

n n Let s(Z, P ) ∈ A∗(Z) be the Segre class of a closed subscheme of P . We n n write its image in A∗(P ) under the canonical map i∗ : A∗(Z) → A∗(P ) in the P n n−m form s(Z, P )mh , where h is the class of a hyperplane. 32 LECTURE 2. INTERSECTION THEORY

Proposition 2.3.1. Let (d0, d1, . . . , dn) be the multi-degree of a Cremona transfor- n mation. Let (X, π, σ) be its resolution and Z be the closed subscheme of P such n n that π : X → P is the blow-up of a closed subscheme Z of P . Then

k k X k−ik n dk = d − d i s(Z, P )n−i. i=1

∗ ∗ Proof. We know that σ OPn (1) = π OPn (d)(−E) = OX0 (dH − E), where ∗ OX (H) = π OPn (1) and E is the exceptional divisor of π. We have h = n c1(OPn (1)) for each copy of P . Thus

k k n−k X i k−ik k−i i n−k dk = π∗(dH − E) · h = ((−1) d i π∗(H · E )) · h i=0

k k X i k−ik k−i i n−k X i k−ik i n−i = (−1) d i h · π∗(E ) · h = (−1) d i · π∗(E ) · h i=0 i=0

k k k X i k−ik i n−i k X k−ik n = d + (−1) d i · π∗(E ) · h = d − d i s(Z, P )n−i. i=1 i=1

Note that one can invert the formulas to express the Segre classes

n n−k s(Z, P )k = skh in terms of dk’s.

n−k−1 X in−k n−k−i sk = (−1) (−1) i d di. (2.17) 0≤i≤n−k

n Deﬁnition 2.3.1. A closed subscheme Z in P is called homaloidal if there exists a homaloidal linear system HX with Bs(HX ) = Z.

Observe that Proposition 2.3.1 gives a necessary condition for the homaloidal linear system n s(Z, P )0 ≡ 1 mod d. (2.18) 2.4. HOMALOIDAL LINEAR SYSTEMS IN THE PLANE 33

2.4 Homaloidal linear systems in the plane

Let b be the base ideal of a homaloidal linear system in the plane. Since |b(d)| = |b(d)|, we may assume that it is integrally closed. It is known that in this case the blow-up scheme Bl(b) is a normal surface. Let X → Bl(b) be its minimal 2 2 resolution of singularities and σ : X → P be the composition X → Bl(b) → P . A birational morphism of nonsingular surfaces is a composition of blow-ups of closed points on smooth surfaces (see [Hartshorne]). Let

σN σN−1 σ2 σ1 2 X = XN / XN−1 / ... / X1 / X0 = P be the sequence of such blow-ups. Here each σi : Xi → Xi−1 is the blow-up of a closed point xi ∈ Xi−1. For any N ≥ b > a ≥ 1, let

σb,a = σa ◦ ... ◦ σb : Xb → Xa−1.

−1 Let Ei = σi (xi) be the exceptional curve of σi. It is a smooth rational curve on Xi with self-intersection equal to −1. We say that a point xj is infinitely near to xi of order 1 and write xj xi if j > i and σj,i+1 : Xj → Xi is an isomorphism in a neighborhood of xj and maps xj to a point in Ei. The Enriques diagram is an oriented graph whose vertices are point x1, . . . , xN and the arrow goes from xj to xi if xj xi. If a point xj is connected to xi by an oriented path of length k, we say that xj is infinitely near to xi of order k and write xj k xi. The points 2 from which no edge exits can be identified with points in P , they are called proper 2 points. However, strictly speaking, only one point lies in P , namely the point x1. Any non-proper point xi ∈ Xi−1 is mapped by σi,1 to some proper point. Let −1 −1 Ei = σN,i(xi) = σN,i−1(Ei), i = 1,...,N. Since we are blowing up points on smooth surfaces, the scheme-theoretical pre- image is a reduced curve. It is reducible if and only if there are no points infinitely j near to it. The irreducible components of Ei consist of proper transforms Fi in X of Ei and Ej, where xj is infinitely near to xi of some order. We have Ej ⊂ Ei j if and only if xi = xj or xj is infinitely near to xi. The self-intersection of Fi is equal to −1 − ai, where ai is the number of points infinitely near to xi of order 1. 2 2 2 Since Ei = −1, we have Ei = Ei = −1. Also Ei ·Ej = 0 if i 6= j. This is because we can always replace the divisor Ei by a linearly equivalent divisor which does not map to xj. ∗ The divisor classes e0 = σ (line), ei = [Ei], i = 1,...,N, form a basis in 2 2 A1(X) = Pic(X) = H (X, Z). It is an orthonormal basis in the sense that e0 = 2 1, ei = −1, i > 0, ei · ej = 0, i 6= j. 34 LECTURE 2. INTERSECTION THEORY

2 Let H = H0 = |b(d)| be a homaloidal linear system of degree d on P . Let m1 be the multiplicity of a general member D at x1. The the pre-image of H0 in X1 has the fixed component m1E1. We subtract it and consider the linear system ∗ H1 = |σ1(H0) − m1E1|. Let m2 be the multiplicity of a general member of H1 at ∗ x2 ∈ X1. Then |σ2(H1) − m2E2| has no fixed components on X2. Continuing in this way, we find that N ∗ X HN = σ (H) − miEi i=1 has no base points and no fixed components. Since it defines a birational morphism 2 to P we must have

N ∗ X 2 X 2 σ (H) − miEi = d − mi = 1. (2.19) i=1

Recall that the behavior of the canonical class of a smooth variety X under the blow-up σ : BlZ X → X of a smooth closed subvariety Z ⊂ X of codimension k

∗ KBlZ X == σ (KX ) + (k − 1)E, (2.20) where E is the exceptional divisor. In our case we obtain X KX = −3H + Ei.

Since a general member of a homaloidal linear system is a rational variety, a general member of the linear system HN is a smooth rational curve C. We have 2 2 C = 1, and, by the adjunction formula C + C · KX = −2, X − 3 = C · KX = −3d + mi. (2.21)

The two equalities (2.19) and (2.21) give necessary conditions for the existence of a homaloidal linear system of plane curves of degree d passing though the points x1, . . . , xN (including infinitely near) with multiplicities m1, . . . , mN . They also sufficient, if we check, that the dimension of such linear system is equal to 23 and the linear system H does not fixed components.

Example 2.4.1. Assume that m1 = ... = mN = m. We have

d2 − Nm2 = 1, 3d − Nm = 3.

3This is the expected number since passing through a point with multiplicity m gives (m+1)m/2 P conditions so that the two equalities give (d + 2)(d + 1)/2 − 1 − mi(mi + 1)/2 = 2. 2.5. SMOOTH HOMALOIDAL LINEAR SYSTEMS 35

The only solutions are

(d, m, N) = (1, 0, 0), (2, 1, 3), (5, 2, 5), (8, 3, 7), (17, 6, 8).

The first transformation is, of course, a projective transformation. The second one is a quadratic transformation defined by the linear system of conics through 3 points. They could be infinitely near, but not collinear (this extends even to infinitely near points meaning that |H − E1 − E2 − E3| = ∅. The third linear system is defined by the linear system of 6-nodal plane quintics. If we assume that the base points are proper and in general position, then the blow-up of the six points is 3 isomorphic to a nonsingular cubic surface in P (by the map defined by the linear system of cubics through the six points). The exceptional curves of the blow-up are mapped to six skew lines on the cubic surface. The six skew lines which form a double-six with the set of the exceptional curves are blown down to six points in 2 P P by the linear system |3H − Ei|. The P -locus of the Cremona transformation consist of the union of 6 conics passing though the six points except one. The last two transformations are more elaborate, they are called the Geiser transformation and the Bertini transformation, respectively. P Remark 2.4.1. The ideal sheaf σ∗(OX (− miEi)) is an integrally closed ideal sheaf in the plane. According to Zariski, any integrally closed ideal in the plane is obtained in this way. For example, consider the ideal sheaf of the subscheme with support at a point p and given in local coordinates by (x, yn). Its blow-up is the scheme isomorphic over an affine neighborhood of p with coordinates x, y n 2 1 to X = Proj C[x, y][u, v]/(uy − vx) ⊂ A × P . In a local chart with v 6= 0, it is nonsingular. In the local chart u 6= 0, it has a singular point locally isomorphic to a singular point (0, 0, 0) on the affine surface vx − yn = 0. The type of this singularity goes under the name An−1-singularity. The exceptional divisor of its minimal resolution X0 → X consists of a chain of n − 1 smooth rational curves with self-intersection −2. The exceptional divisor of the composition 0 2 X → X → P consists of a chain of n smooth rational curves, with a (−1)-curve at one end, and all other curves are (−2)-curves. These curves form an exceptional configuration E1 obtained by a sequence of blowing ups of infinitely near points xn xn−1 ... x1 = x.

2.5 Smooth homaloidal linear systems

Deﬁnition 2.5.1. A homaloidal linear system H is called smooth if the reduced base scheme is smooth and the base ideal b = b(H) is a reduction of the product of the powers of the radicals of the primary components of b. 36 LECTURE 2. INTERSECTION THEORY

√ r If bi are the primary components of b and bi is a reduction of ( bi) i for some ri ≥ 1, we write k X H = |dH − riZi| i=1 √ where H is meant to be the divisor class√ of a hyperplane and Zi = V ( bi). Since each Zi is smooth, the ideals bi and their powers are integrally√ closed. Also the blow-up scheme X of the product of the ri-th powers of bi is smooth P ∼ n b with exceptional divisor equal to riEi. Each divisor Ei = P(NZi/P ). Since becomes invertible on X, X is a smooth resolution of any Cremona transformation f deﬁned by the homaloidal linear system. Note that the ideal b is not necessary integrally closed, so the blow-up of b may 2 2 be non-normal. For example, a primary component bi may look like (x , y ) with integral closure equal to (x2, y2, xy). Let |dH − r1Z1 − ... − rN ZN | be a smooth homaloidal linear system. Let P E = riEi be the exceptional divisor of the integral closure of b(HX ). We have

N N n n−k−1 n−k X n−k−1 n−k X n−k n s(Z, P )k = (−1) π∗(E ) = (−ri) π∗(Ei ) = ri s(Zi, P )k. i=1 i=1 (2.22)

n We have already explained how to compute s(Y, P ) for any smooth subvariety n n Y of P . For example, if C is a curve of degree d in P , we get n −1 s(C, P ) = c(NC/Pn ) = 1 − c1(NC/Pn ). The exact sequence of sheaves of differentials

∨ ∗ 1 0 → N n → j Ω n → Ω → 0 C/P P C gives an isomorphism

n n−1 ∗ ^ ∼ ^ ∨ 1 j Ω n N n ⊗ Ω . P = C/P C This yields

c1(NC/Pn ) = −KPn · C + 2g − 2 = (n + 1) deg C + 2g − 2, where g is the genus of C. Hence  2 − 2g − (n + 1) deg C if k = 0, n  s(C, P )k = 1 if k = 1,  0 otherwise. 2.5. SMOOTH HOMALOIDAL LINEAR SYSTEMS 37

So, if we assume that Bs(HX )red consists of M curves C1,...,CM and N points x1, . . . , xN , formula (3.3) together with Proposition 2.3.1 give PM PN Proposition 2.5.1. Assume that HX = |dH − i=1 riCi − i=1 mixi|, where Ci, i = 1,...,M, is a smooth curve of degree deg Ci and genus gi, and xi are isolated points. Then

M N n X n n−1 n X n dn = d + [(ri (n + 1) − dnri ) deg Ci + ri (2gi − 2)] − mi = 1 i=1 i=1 M n−1 X dn−1 = d − deg Ci, i=1 k dk = d , k = 0, . . . , n − 2. To ﬁnd the dimension of a smooth homaloidal system we use some known n techniques. Let J be the ideal sheaf of a smooth closed subscheme Z of P . We have a sequence of ideals J m ⊂ J m−1 ⊂ ... ⊂ J with quotients J k/J k+1 isomorphic to the k-symmetric powers of N ∨ = Y/Pn J /J 2. We also use the exact sequence

m 0 → (J /J )(d) → OmZ (d) → OZ (d) → 0, m where mZ = V (JZ ). The exact sequence gives 0 0 m h (OmZ (d)) = OZ (d) + h (J /J (d)), 1 m provided we have checked that h (J /J (d)) = 0. Suppose we know c(NY/Pn ). Then, one can compute the Chern class of the symmetric powers of N ∨ , also Z/Pn of its twists N ∨ (d), and then apply the Riemann-Roch Theorem to compute Z/Pn 0 m OZ (d) and h ((J /J )(d)). Example 2.5.1. Assume that Y is a smooth curve C of genus g and degree deg C. We have c (Sk(N ∨ )) = 1 k(k + 1)c (N ∨ ) and by, the additivity of c , we 1 C/X 2 1 C/Pn 1 know m−1 m 1 X m+1 c1(J /J ) = 2 k(k + 1)c1(NC/Pn ) = 3 c1(NC/Pn ). k=1 Next, we use that, for any locally free sheaf E of rank r, and an invertible sheaf L, we have r X i c(E ⊗ L) = cr−i(E)(1 + c1(L)) . (2.23) i=0 38 LECTURE 2. INTERSECTION THEORY

Finally, we use Riemann-Roch Theorem which allows us to compute the Euler n m characteristic χ(P , J /J (d)). m 1 Since the rank of J /J is equal to 2 + ... + m = 2 (m − 1)(m + 2), we obtain

c (J /J m(d)) = m+1c (N ∨ ) + 1 (m − 1)(m + 2)d deg C. 1 3 1 C/P3 2 By Riemann-Roch, for any locally free sheaf E of rank r,

χ(E) = deg c1(E) + r(1 − g). This gives

m m+1 1 1 3 χ(J /J (d)) = 3 c1(NC/P )+ 2 (m−1)(m+2)d deg C + 2 (m−1)(m+2)(1−g). m m χ(OmC (d)) = χ(J /J (d)) + χ(OC (d)) = χ(J /J (d)) + d deg C + 1 − g m+1 m+1 = 3 (−4 deg C + 2 − 2g) + 2 (d deg C + 1 − g) m+1 m+1 m+1 m+1 = (−4 3 + d 2 ) deg C + (2 3 + 2 )(1 − g). Applying (2.5.1), we have

d3 − 1 = (−4m3 + 3dm2) deg C + 2m3(1 − g). (2.24)

Observe that this implies that d is an odd number. Taking (2.24) into account, we can rewrite the previous equality in the form 1 χ(O (d)) = (d3 − 1 + m(4 + 3d) deg C + m(3m + 1)(1 − g)). mC 6

0 m Finally, under assumption, that χ(OC (d)) = h (OC (d)) and χ(J /J (d)) = h0(J /J m(d)), we get

0 m d+3 4 = h (J (d)) = 3 − χ(OmC (d)) 1 = (6d2 + 11d + 7 − m(4 + 3d) deg C + m(3m + 1)(g − 1)). (2.25) 6 The equalities (2.24) and (2.25) give necessary conditions for the existence of a homaloidal linear system |dH − mC|. Remark 2.5.1. When n > 2, there are no smooth homaloidal linear systems with P 0-dimensional base locus. Indeed, suppose such a linear system |dH − rixi| exists. We know that the multi-degree is equal to (d, d2, . . . , dn). Let S be the pre-image of a general plane. This is a surface of degree dn−2. The restriction P of the linear system to S is of the form |dh − rixi|, where h is a hyperplane 2.5. SMOOTH HOMALOIDAL LINEAR SYSTEMS 39 section of S. After we blow-up the base points, we get a degree 1 map onto a 2 2 P 2 n P 2 plane. This implies that 1 = d h − mi = d − mi . On the other hand, we n P n P n P 2 have 1 = d − mi . This gives ri = mi , hence all mi are equal to 1, and the base scheme is reduced and consists of N = dn − 1 points. We know that the n−1 pre-image of a general line is an irreducible curve R of degree dn−1 = d . The linear subsystem of HX that consists of divisors passing through R is of dimension n − 2 (it is isomorphic to the linear system of hyperplanes through a line in the target space). Take two general points on R, then we ﬁnd a divisor from HX that passes through these two points and also dn − 1 base points. Thus it intersects R at dn + 1 > dn points contradicting the Bezout Theorem. Example 2.5.2. Let n = 3. We have

M N 3 X 3 2 3 X 3 1 = d + [(4ri − 3dri ) deg Ci + ri (2gi − 2)] − mi . (2.26) i=1 i=1 Take d = 2. We have M N X 3 2 3 X 3 1 = 8 + [(4ri − 6ri ) deg Ci + ri (2gi − 2)] − mi . (2.27) i=1 i=1 This leaves us with a few possibilities. By the previous remark, M 6= 0. Since each quadric from the linear system passes through Ci with multiplicity ri, and any the intersection of two quadrics is a connected curve of degree 4, we must have ri ≤ 2 and X ri deg Ci < 4.

If deg C1 = 3, then M = 1, r1 = 1, g1 = 0, and (2.27) gives 1 = 8 − 6 − 2 − PN 3 2 deg C2 − i=1 mi , a contradiction. If deg C1 = 2, then r1 = 1, g1 = 0. If M = 2, then r2 = 1, deg C2 = 1, g2 = PN 3 0, and we obtain 1 = 8 − 4 − 2 − 2 − i=1 mi , a contradiction. If M = 1, we PN 3 get 1 = 8 − 4 − 2 − i=1 mi . This gives N = 1, m1 = 1. This case is realized by the linear system of quadrics through a conic and a point lying outside of the plane spanning the conic. The multi-degree is (2, 2). We will discuss these and more general quadro-quadratic transformations of this kind in the next lecture. If deg C1 = 1, then r1 ≤ 2, g1 = 0. If r1 = 2, then the quadrics must be of the 2 2 form V (aL1 + bL1L2 + cL2), where C1 is the intersection of hyperplanes V (L1) and V (L2). The dimension of such system is equal to 2, so it is not homaloidal. If M = 2, for the same reason, r2 = 1and deg C2 = 1. We get 1 = 8 − 2 − PN 3 2 − 2 − i=1 mi , a contradiction. So, the remaining possibility is M = 1, and N = 3, m1 = m2 = m3 = 1. This case is realized and leads to a Cremona transformation of multi-degree (2, 3). 40 LECTURE 2. INTERSECTION THEORY

Example 2.5.3. It is possible that Bs(T ) is 0-dimensional but the homaloidal linear system is not smooth. There are hidden 1-dimensional inﬁnitely near com- 3 ponents. For example, consider the linear system of quadrics in P which pass through 3 non-collinear points and have the same tangent plane at the fourth point. Choosing coordinates, we may assume that the points are p1 = [1, 0, 0, 0], p2 = [0, 1, 0, 0], p3 = [0, 0, 1, 0], p4 = [0, 0, 0, 1], and the tangent plane at the point p4 has equation t0 + t1 + t2 = 0. After we blow-up the ﬁrst three points, we obtain that the inverse image of the linear system has the base locus equal to a line in the exceptional divisor E4 over the point p4. If we blow-up this line, we resolve 0 the birational map. The exceptional divisor consists of E1,E2,E3,E4,E5, where 0 2 E4 is the proper transform of E4 isomorphic to P and and E5 is the exceptional 0 divisor over the line isomorphic to the minimal ruled surface F2. The divisors E4 and E5 intersect along a curve C which is the exceptional section in E5 and a line 2 in P . The base ideal in a neighborhood of the point p4 is isomorphic to the ideal (xy, yz, xz, x + y + z). After we make the change of variables x → x + y + z, it becomes isomorphic to the ideal (x, y2, yz, z2). It is easy to see that the blow-up scheme is isomorphic to the projective cone over the blow-up of the maximal ideal (y, z). It has a singular point locally isomorphic to the cone over the Veronese surface. Its exceptional divisor is isomorphic to the quadratic cone. The birational morphism from the resolution in above to the blow-up of the base scheme is the 0 contraction of the divisor E4 to the singular point of the blow-up. So the homaloidal linear system can be written in the form |2H −p1 −p2 −p3 − 2 2 Z4|, where p4 = (Z4)red, and Z4 is locally given by a primary ideal (x, y , yz, y ).

2.6 Special Cremona transformations

A Cremona transformation with smooth connected base scheme is called special Cremona transformation. There are no such transformations in the plane and they are rather rare in higher-dimensional spaces and maybe classiﬁable. We start from one-dimensional base schemes. It follows from our formulas that the following two examples work:

(i) n = 3, g = 3, deg C = 6, d = 3;

(ii) n = 4, g = 1, deg C = 5, d = 2.

The ﬁrst example is a bilinear (or a cubo-cubic transformation) which we will study in detail later. The second transformation is given by quadrics. The restriction of the homaloidal linear system to a general hyperplane is the linear system of 2.6. SPECIAL CREMONA TRANSFORMATIONS 41 quadrics through 5 points. It is known (see, for example, [Topics]) that the image is a ten-nodal Segre cubic. So, the degree of the inverse transformation is equal to 3. The image of a general plane is a Veronese surface of degree 4. So, the multidegree of the transformation is equal to (2, 4, 3). This is an example of a 4 quadro-cubic transformation in P discussed in Semple-Roth’s book. Note that the base locus of the inverse transformation f −1 is a surface of degree 5, an elliptic scroll. In fact, the image of a general plane under f is of degree 4 but given by cubics. This implies that the plane meets the base scheme at 5 simple points. A result of B. Crauder and S. Katz shows that the two cases are the only possible cases with a smooth connected one-dimensional locus. Next we assume that the dimension of the base locus is equal to 2. The next result also belongs to Crauder and Katz.

Theorem 2.6.1. A special Cremona transformation with two-dimensional base scheme Z is one of the following:

(i) n = 4, d = 3,Z is an elliptic scroll of degree 5, the base scheme of the inverse of the quadro-cubic transformation from above;

(ii) n = 4, d = 4,Z is a determinantal variety of degree 10 given by 4×4-minors of a 4 × 5-matrix of linear forms (a bilinear transformation, see later).

(iii) n = 5, d = 2,Z is a Veronese surface.

(iv) n = 6, d = 2,Z is an elliptic scroll of degree 7;

(v) n = 6, d = 2,Z is an octic surface, the image of a plane under a rational map given by the linear system of quartics through 8 points.

In cases (ii) and (iii) the inverse transformation is similar, with isomorphic base scheme. There is no classiﬁcation for higher-dimensional Z. However, we have the following nice results of L. Ein and N. Shepherd-Barron [Amer. J. Math. 1989]. n 1 Recall that a Severi variety is a closed a subvariety Zof P of dimension 3 (2n− n+1 4) such that the secant variety is a proper subvariety of P . All such varieties are classiﬁed by F. Zak.

5 (i) Z is a Veronese surface in P ; 5 14 (ii) Z is the Grassmann variety G1(P ) embedded in the Plucker¨ space P ; 2 2 8 (iii) Z is the Severi variety s(P × P ) ⊂ P ; 26 (iv) Z is the E6-variety, a 16-dimensional homogeneous variety in P . 42 LECTURE 2. INTERSECTION THEORY

Theorem 2.6.2. Let f be a quadro-quadratic transformation with smooth base scheme. Then the base scheme is isomorphic to one of the Severi varieties.

All these transformations are involutions (i.e. T = T −1). In particular, their multi-degree vectors are symmetric.

2.7 Double structures

Let Y ⊂ X be a smooth closed codimension r subvariety of a n-dimensional ∨ 2 smooth varierty X and NY/X = JY /JY be its conormal locally free sheaf. A ∨ double structure on Y is a surjection u : NY/X → L onto an invertible sheaf L. ∨ It defines a section s : Y → P(NY/X ). Let u˜ : JY → L be the composition of 2 u and the canonical surjection JY → JY /JY . Its kernel is an ideal sheaf I that defines a closed subscheme Z containing Y as a closed subscheme and contained 2 in the first infinitesimal neighborhood Y (1) defined by the ideal sheaf JY . Thus, by definition, we have an exact sequence:

0 → JZ → JY → L → 0. (2.28)

We call the scheme Z a double structure on Y . In the case when X is a smooth surface, and Y is a closed point, the double structure is a choice of a point on the exceptional curve E of the blow-up of the point. It corresponds to a tangent direction at the point. It is an infinitely near point to Y . Let E1 be the exceptional divisor of the blow-up σ1 : B1 = BlY X → X and Y1 be the image of the section s : Y → E1 = P(NY/X ) defined by L. The exact k sequence (2.28) gives a surjection of graded algebras ⊕JY → Sym L. Its kernel −1 ∨ is generated by σ1 JZ . Recall that the affine cone CE over E is Spec Sym NY/X . ∨ The section s corresponds to the surjection Sym NY/X → Sym L. This shows that ∨ the pre-image of the ideal sheaf of Y1 in CE is generated by Ker(u) ⊂ NY/X . This −1 also shows that the ideal JY1 of Y1 in E is equal to σ1 (JZ ) ·OE1 and hence

−1 σ1 (JZ ) = OB1 (−E1) ·JY1 . (2.29)

The quotient OB1 (−E1)/OB1 (−E1) ·JY1 is isomorphic to OY1 (−E1). It follows ∗ ∼ from the deﬁnition of OE(1) = OE(−E1) that s (OE(−E1)) = L. This shows that we have an exact sequence

0 → OB1 (−E1) ·JY1 → OB1 (−E1) → L → 0.

It is obtained from (2.28) by applying σ−1. 2.7. DOUBLE STRUCTURES 43

Let σ2 : B2 → B1 be the blow-up of Y1. Then the inverse transform of JZ in 0 ∗ B2 becomes invertible and isomorphic to OB2 (−E1 − E2), where E1 = σ2(E1) is the full transform of E1 in B2 and E2 is the exceptional divisor of σ2. We have

n−m−1 n−m s(Z,X)m = (−1) σ∗(E ), where E = E1 + E2 and σ = σ1 ◦ σ2 : B2 → X. By the projection formula,

n−m n−m X n−m i n−m−i σ2∗(E1 + E2) = i σ2∗(E1 · E2 ) i=0

n−m n−m X n−m i n−m−i X n−m i n−m−i−1 = i E1 · σ2∗(E2 ) = i E1(−1) s(Y1,B1)m+i. i=0 i=0 Thus

n−m−1 X n−m i s(Z,X)m = s(Y,X)m + i (−E1) s(Y1,B1)m+i. (2.30) i=0

So we need to compute s(Y1,B1) = c(NY1/B1 ). We use the following well- known result about projective bundles.

Lemma 2.7.1. Let E be a locally free sheaf of rank r on a smooth scheme X and let P(E) be the projective bundle associated to E. Let u : E → L be a surjection of E onto an invertible sheaf L and s : X = Proj Sym L → P(E) = Proj Sym E is the corresponding section. Then

∗ ∼ ∨ s (Ns(X)/P(E)) = Ker(u) ⊗ L.

Proof. Let P = P(E) and S = s(X). Consider the exact sequence of sheaves of relative differentials

∨ 1 1 0 → NS/P → ΩP/X ⊗ OS → ΩS/P → 0.

Since the projection p : P → X induces an isomorphism S → X, we have 1 ΩS/P = 0. Thus ∨ ∼ 1 NS/P = ΩP/X ⊗ OS. (2.31) Now consider the Euler exact sequence

∗ ∨ 1 ∨ 0 → OP → p (E ) ⊗ OP (1) → (ΩP/X ) → 0. (2.32) 44 LECTURE 2. INTERSECTION THEORY

∼ ∗ It follows from the deﬁnition of the sheaf OP (1) that L = s (OP (1)). Passing to the duals, and applying s∗, we obtain

∗ ∨ ∼ 1 ∼ −1 ∼ −1 s (NS/P ) = ΩP/X = Ker(E ⊗ L → OX ) = Ker(u) ⊗ L . Taking the dual again, we get

∗ ∼ ∨ s (NS/P ) = Ker(u) ⊗ L. (2.33)

Note that equalities (2.31) and (2.32) give

1 ∨ ∨ c(NS/P ) = c((ΩP/X ) ) = c(E ⊗ L). (2.34) Applying (2.23) and (2.31), we ﬁnd

r X i c(NY1/E) = cr−i(NY/X )(1 + D) . (2.35) i=0

Since OB(1) = OB(−E), we get

−1 NE/B|Y1 = OE(E)|Y1 = L = OY (−D). (2.36)

Using exact sequence (2.12), we ﬁnally obtain

r X r−i −1 −1 s(Y1,B) = ( ci(NY/X )(1 + D) ) (1 − D) . i=0 and, applying (2.30) and (2.36), we get

n−m−1 X n−m j s(Z,X)m = s(Y,X)m + j D · s(Y1,B)m+j. (2.37) j=0

Example 2.7.1. Let Y be a closed point. We have s(Y,X)0 = s(Y1,B)0 = [Y ], so

s(Z,X) = 2[Y ].

Or let Y be a smooth curve on a n-fold X. Then

−1 s(Y,X) = ([Y ] + c1(NY/X )) = [Y ] − c1(NY/X ),

−1 −1 s(Y1,B) = ([Y ] + (n − 1)D + c1(NY/X )) (1 − D) 2.7. DOUBLE STRUCTURES 45

= [Y ] − (n − 1)D − c1(NY/X ))(1 + D)

= [Y ] − (n − 2)D − c1(NY/X ). This gives

s(Z,X)0 = −2c1(NY/X ) + 2D (2.38)

s(Z,X)1 = 2[Y ].

n Let us apply this to our situation when X = P and Z is the base scheme of a homaloidal linear system. We have

k k X k k−i n dk = d − i d s(Z, P )n−i. i=1 n Example 2.7.2. Let Y be a curve C of genus g in P . Consider a double structure on C deﬁned by an invertible sheaf O(D) of degree a = deg D. From the previous computations c1(NC/Pn ) = 2g − 2 + (n + 1) deg C. Applying (2.38), we get n n s(Z, P )0 = 4 − 4g − 2(n + 1) deg C + 2a, s(Z, P )1 = 2 deg Y, hence

n dn = d − (4 − 4g − 2(n + 1) deg C + 2a) − 2nd deg C, (2.39) n−1 dn−1 = d − 2 deg C, k dk = d , k < n − 1.

Consider a special case when n = 3 and C is a line. By taking d = 2, a = −1, we obtain d3 = 2. Thus adding one isolated base point, we obtain a candidate for a homaloidal linear system of quadrics through the double structure of a line and an isolated point. We will see later that this corresponds to a degeneration of 3 a quadratic transformation through a smooth conic in P and a point outside the plane spanned by the conic. 3 Another special case when C is a twisted cubic in P and a = −4. We take d = 3, and get d3 = 1. We will see later that this is a special case of a cubo-cubic Cremona transformation.

By considering a double structure on the section Y1 of the blow-up of Y , we can introduce a triple structure on Y , and continuing in this way, we can deﬁne a k-multiple structure on Y . It is given by a sequence of the blow-ups of smooth closed subvarieties Yi ⊂ Bi−1

σk σk−1 σ2 σ1 B = Bk / Bk−1 / ... / B1 / B0 = X 46 LECTURE 2. INTERSECTION THEORY such that Y0 = Y , and Yi, i > 0, is a section of the exceptional divisor Ei → Yi−1 of σi deﬁned by a surjection NYi−1/Bi−1 → Li, where Li is an invertible sheaf on Yi−1. Note that the restriction of the composition σ1 ◦ ... ◦ σi : Bi → B0 = X to ∼ Yi deﬁnes an isomorphism Yi = Y . So we can identify Li with an invertible sheaf on Y . For any k ≥ j ≥ i ≥ 1, let ( σi ◦ ... ◦ σj : Bj → Bi−1 if j > i, σj,i = σj if j = i.

Also set −1 Ej,i = σj,i (Yi−1).

Consider the inclusion of invertible sheaves on Bk

k k−1 X X OBk (− Eki) ⊂ OBk (− Eki) ⊂ ... ⊂ OBk (−Ek1 − Ek2) ⊂ OBk (−Ek1). i=1 i=1 Pj Let Zj be the subscheme of X deﬁned by the ideal sheaf (σk1)∗OBk (− i=1 Eki). Then we have a chain of closed subschemes

Y = Z1 ⊂ Z2 ⊂ ... ⊂ Zk (2.40) ∼ with JZj /JZj+1 = Lj, j = 1, . . . , k − 1. By the projection formula,

j j X X (σk1)∗OBk (− Eki) = (σj1)∗(− Eji). i=1 i=1

Thus, each Zj is a j-th multiple structure on Y . Finally, we may consider the scheme

k X Zk(m1, . . . , mk) = V ((σk1)∗OBk (− miEki)). i=1

Pk Its proper transform on Bk is the linear system |dH − i=1 miEki|. For example, Zk = Zk(1,..., 1). By deﬁnition, mZk = Zk(m, . . . , m). Generalizing the computations for the Segre class of a double structure given in (2.30), we can compute, by induction, the Segre classes of the base scheme Z of Pj |dH − i=1 miEki|. We get

n−c−1 n−c X n−c j n−c−j j s(Zk(m1, . . . , mk),X)c = m1 s(Y,X)c + j m1m2 D1s(Y1,B1)c+j j=0 2.7. DOUBLE STRUCTURES 47

n−c−1 X n−c j n−c−j j + j m2m3 D2s(Y2,B2)c+j + ... + j=0

n−c−1 X n−c j n−c−j j j mk−1mk Dk−1s(Yk−1,Bk−1)c+j. j=0

where s(Yj,Bj) are computed by induction using (2.12),

r X r−i −1 −1 s(Yj,Bj) = ( ci(NYj−1/Bj−1 )(1 + Dj) ) (1 − Dj) . (2.41) i=0

Example 2.7.3. Assume that the base scheme is a k-multiple structure Zk on a smooth curve C. Applying (2.41), we ﬁnd

s(Yj,Bj)0 = −(n − 2)Dj + s(Yj−1,Bj−1)0 =

[Yj] − (n − 2)Dj − (n − 2)Dj−1 − s(Yj−2,Bj−2)0

= −(n − 2)(D1 + ... + Dj) − c1(NC/Pn ), This gives

k−1 k−1 n X n X n−1 s(Zk(m1, . . . , mk), P )0 = mj s(Yj,Bj)0 + n mjmj+1 Dj, j=1 j=1 k n X n−1 s(Zk(m1, . . . , mk), P )1 = mj [C]. j=1

For a concrete example, let us take all mj = m. Then we obtain

k−1 k−1 n n X X s(mZk, P )0 = −m [(n − 2)( (k − i)Di) + kc1(NC/Pn ) − n Di], i=1 i=1 n n−1 s(Zk, P )1 = km [C].

As soon as we know the Segre classes of Zk(m1, . . . , mk) we can compute the multi-degrees of the transformation and check whether dn = 1, the necessary condition for a homaloidal linear system. Another condition to check is dim |dH − Zk(m1, . . . , mk)| = n. For this we use the Riemann-Roch Theorem (see [Hartshorne], Appendix]. We state it only in the case n = 3. 1 1 χ(X, O (D)) = (2D3−3K ·D2+D·K2 )+ D·c (Ω1 )+χ(O ). (2.42) X 12 X X 12 2 X X 48 LECTURE 2. INTERSECTION THEORY

Pk In our situation, X = Bk,D = dH − E, where E = j=1 mjEkj,

k X KX = −4H + Eki. i=1

1 We also use the following formula for c2(X) := c2(ΩX ) from [Fulton], Example 15.4.3. It is obtained by successive application of the formula

∗ 2 c2(Bj) = σj (c2(Bj−1)) + (2 − 2g)fj − Ej , (2.43) where fj is the class of a ﬁbre of Ej → Yj−1. We skip the computations which show that k X c2(Bk) · D = 24d + (d − 4) deg C( mj. j=1

Note that c2(Bk) · D = 24 if d = 4 . This reﬂects the fact that, for any nonsingular 3-fold X

c2(X) · c1(X) = 24χ(X, OX ). Pk Note that c1(Bk) = 4H − j=1 Ekj. To compute the other ingredients in the Riemann-Roch formula, we use that, for any i > j,

Eki ·Ekj = 0 because, replacing Ej by an algebraically equivalent cycle, we can assume that it ∗ 3 does not intersect Yj−1, hence σki(Ej) ·Ekj0 must be zero. Also we have Ekj =

−c1(NYj−1/Bj−1 ). Using this we obtain

k k 3 3 X 3 X 2 D = d + mj c1(NYj−1/Bj−1 ) − 3d deg C mj , j=1 j=1 k k 2 2 X 2 X 2 KBk · D = −4d + (4mi + 2dmi) deg C − mj c1(NYj−1/Bj−1 ), j=1 j=1 k k X X K2 · D = 16d − (d + 8m ) deg C + m c (N ), Bk i j 1 Yj−1/Bj−1 j=1 j=1 k X c2(Bk) · D = 6d + (d − 4) deg C mj. j=1 2.8. DILATED TRANSFORMATIONS 49

Finally, we get

k d+3 X 3 2 12χ(Bk, OX (dH − E)) = 12 3 + (2mj + 3mj + mj)c1(NYj−1/Bj−1 ) j=1

k k X 2 X −((6d + 12) mj + (5d + 12) mj + kd) deg C. j=1 j=1 If |dH − E| is a homaloidal system, the right-hand side must be equal to 36 (pro- 1 vided that H (Bk, OBk (dH − E)) = 0). To convince the reader that we have not made a mistake let us make a few tests. First, we take deg C = 2, k = m1 = 1. The right-hand side is equal to 120 + 36 − 2(24 + 22 + 2) = 60. Since dim |2H − C| = 4, the formula is correct. Next, we take deg C = 3, k = 2, deg D1 = −4, m1 = m2 = 1. The right- hand side is equal to 54 + 108 + 66 + 6(10 + 18) − 6(30 + 27 + 3) = 36. We know that the linear system is homaloidal, so the formula is correct. 3 The conditions (dH − E) = 1 and χ(Bk, OX (dH − E)) = 4 are not the only necessary conditions for the existence of a homaloidal linear system. Note that a general member D of |dH − E| is a smooth rational surface. This implies that its canonical linear system is empty. By the adjunction formula,

k X KD = ((d − 4)H − (mi − 1)Ekj) · D. j=1

For example, if all mj = 1, this implies that d < 4. We have seen already an example with d = 3, k = 2 and deg C = 3. One more possible example is d = 3, k = 3, deg C = 2 and the triple multiple structure Z on the conic is deﬁned by line bundles of degree −2 and −3. The arithmetic genus of Z is equal to 3, the same as the genus of the double structure on a twisted cubic. In the next lecture we will study cubo-cubic transformations with basis scheme an arbitrary arithmetical Cohern-Macaulay one-dimensional schme of degree 6 and arithmetic genus 3. Our examples are special cases of such transformations.

2.8 Dilated transformations

n−1 Starting from a Cremona transformation T in P we seek to extend it to a Cre- n n n−1 mona transformation in P . More precisely, if po : P 99K P is a projection n n−1 map from a point o, we want to ﬁnd a Cremona transformation T : P 99K P such that po ◦ T¯ = T ◦ po. Suppose that T is given by a sequence of degree d 50 LECTURE 2. INTERSECTION THEORY homogeneous polynomials (G1,...,Gn). Composing with a projective transfor- n mation in P , we may assume that o = [1, 0,..., 0]. Thus the transformation T must be given by (F0, QG1, . . . , QGn), where Q and F0 are coprime polynomials of degrees r and d + r.

Proposition 2.8.1. (I. Pan) Let (G1,...,Gn) be homogeneous polynomials of degree d in t1, . . . , tn. Let F0 = t0A1 + A2,Q = t0B1 + B2, where A1,A2,B1,B2 are homogeneous polynomials of degrees d + r − 1, d + r, r, r − 1, respectively. Assume that F0 and Q are coprime and A1B2 6= A2B1. Then the polynomi- n als (F0, QG1, . . . , QGn) define a Cremona transformation of P if and only if n−1 (G1,...,Gn) define a Cremona transformation of P . Proof. We will give two proofs, a geometric and purely algebraic one. The first one is geometric. Assume that the conditions are satisfied. Take a general point y in the target space. The linear system of hyperplanes through y contains a codimension 1 subsystem of hyperplanes passing through the point o. Thus we can write y as the intersection of n + 1 hyperplanes H1,...,Hn,Hn+1, where H1,...,Hn contains o and Hn+1 does not. The pre-image of y under T is equal to the intersection of n hypersurfaces D1,...,Dn from the linear span of the polynomials QG1, . . . , QGn P and a hypersurface D = V (F0 + aiQGi). The intersection of the first n hypersurfaces is the union of the line ` = hx, oi and the hypersurface V (Q), where x ∈ V (t0) is the unique pre-image of y under T . The intersection D ∩ (V (Q) ∪ `) consists of the union of D ∩ V (Q) and D ∩ `. The first part belongs to the base locus of T . The conditions on the multiplicities of Q and F0 imply that D has a singular point of multiplicity d + r − 1, one less than its degree. Thus the second intersection consists of one point outside the base point. It is easy to see that the conditions are necessary for this. 0 Our second proof is algebraic. Let F (z1, . . . , zn) denote the dehomogeniza- tion of a homogeneous polynomial F (t0, . . . , tn) in the variable t1. It is obvious that (F0,...,Fn) define a Cremona transformation if and only if the field

0 0 0 0 C(F1/F0,...,Fn/F0) := C(F1/F0,...,Fn/F0) = C(z1, . . . , zn).

Consider the ratio F /QG = t0A1+A2 . Dehomogenizing with respect to t , 0 1 t0GB1+GB2 1 we can write the ratio in the form az1+b , where a, b, c, d ∈ (z , . . . , z ). By our cz1+d C 2 n assumption, ad − bc 6= 0. Then

C(F1/F0,...,Fn/F0) = C(F0/QG1,G2/G1,...,Gn/G1)

az1 + b = C(G2/G1,...,Gn/G1)(F0/QG1) = C(G2/G1,...,Gn/G1)( ). cz1 + d 2.8. DILATED TRANSFORMATIONS 51

This ﬁeld coincides with C(z1, . . . , zn) if and only if C(G2/G1,...,Gn/G1) coincides with C(z2, . . . , zn).

n Deﬁnition 2.8.1. The Cremona transformation T¯ in P obtained from a Cremona n−1 transfromation T in P by the above construction is called a dilation of T . It depends on the choice of a point o and polynomials F0,Q of degrees d + r and r satisfying the conditions on their multiplicities at o stated in the assertion of the previous proposition.

n Deﬁnition 2.8.2. The Cremona transformation T¯ in P obtained from a Cremona n−1 transfromation T in P by the above construction is said to be dilatated from T . It depends on the choice of a point o and polynomials F0,Q of degrees d + r and r satisfying the conditions on their multiplicities at o stated in the assertion of the previous proposition.

It is easy to determine the P-locus of the dilated transformation T¯. Take a line through the point o with parametric equation [s, ta1, . . . , tsn]. Plugging in these equations in the polynomials deﬁning T , we obtain that the image of the line is spanned by the points

[A¯1, B¯1G¯1, B¯1G¯2,..., B¯1G¯n], [A¯2, B¯2G¯1, B¯2G¯2,..., B¯2G¯n], where the bar means that we plug in the coordinates (a1, . . . , an) in the polynomials. It is one point if and only if A¯1B¯2 − A¯2B¯2 = 0. In other words if [a1, . . . , an] is contained in the hypersurface V (A1B2 − A2B1) of degree d + 2r − 1. This hypersurface is a part of the P-locus of T¯. The other part is the the hypersurface V (Q) of degree r which is blown down to the point o. There is one more part that consists of the join of the point o with the P-locus of T in the hyperplane t0 = 0. Of course, some of the parts may overlap. n For example, the standard Cremona transformation Tst in P is given by the formula [t0, . . . , tn] 7→ [t1 ··· tn, t0t2 ··· tn, . . . , t0t1 ··· tn−1].

This is the dilation of the standard Cremona transformation in the plane tn = 0 (the identity if n = 2). Here d = 1, r = 1,F = t0 ··· tn−1,Q = tn,A1B2 − A2B1 = −A2B1 = −t0 ··· tn−1. The P-locus consists of V (t0 ··· tn−1) and V (tn) and consists of n + 1 coordinate hyperplanes, as it should be. The first geometric proof in Proposition 2.8.1 gives some information about the possible multi-degree (d˜1,..., dñ−1) of the dilated transformation. Let (d1,..., n dn−2) be the multi-degree of T . Let L be a general linear subspace of P of dimension k. We can write L as the intersection of n−k−1 hyperplanes H1,...,Hn−k−1 52 LECTURE 2. INTERSECTION THEORY containing L and a hyperplane Hn−k containing L but not containing o. Let Di be the divisors of the homaloidal linear system defining T¯ which corresponds to Hi. The intersection D1 ∩ ... ∩ Dn−k−1 is equal to the union of the cone over the base −1 scheme of T and the cone over CX over a k-dimensional variety T (L ∩ V (t0)) of degree dn−k−1. The intersection D1 ∩ ... ∩ Dn−k−1 ∩ Dn−k is equal to the union of the base locus of T¯ and the intersection of X with Dn−k. The latter is of degree (d + r)dn−k−1. Thus we obtain ˜ dn−k ≤ (d + r)dn−k−1. (2.44)

When k = n − 1, we have the equality. For smaller k, the equality occurs when D∩CX has no components of dimension k belonging to the base scheme of T . It is easy to see that the latter consists of the union of V (F0)∩V (Q) and V (F0)∩CBs(T ). Example 2.8.1. Let n = 3 and T be deﬁned by a smooth homaloidal linear system PN |dh − i=1 mipi|. Let F0 = t0A1 + A2 and Q = t0B1 + B2 as in Proposition 2.8.1 and let ki be the minimal of the multiplicities of A1 and A2 at the point pi and similar ni be for B1,B2. This implies that V (F0) pass through the lines ho, pii with multiplicity ≥ ki and V (Q) passes through these lines with multiplities ni. Let si = min{mi + ni, ki}. Then the dilated homaloidal linear system contains the lines ho, pii with multiplicities si, we obtain X d˜2 = (d + r)d − si.

In a special case, take T to be the standard Cremona transformation Tst given by (t2t3, t1t3, t1t2) and F0 = t0t1. Thus two of the numbers si, corresponding to the base points (0, 1, 0) and (0, 0, 1) are equal to one, and one is equal to 0. We obtain d˜2 = 2. On the other hand, if we take F0 = t0t1 + t2, we obtain d˜2 = 3 and, if we take F0 = t0(t1 + t2), we get d˜2 = 4. Lecture 3

First examples

3.1 Quadro-quadric transformations

Let us show that any vector (2,..., 2) is realized as the multi-degree of a Cremona transformation. For n = 2, we take the homaloidal linear system of conics through three non-collinear points. For n = 3, we can take the homaloidal linear system of quadrics through a nonsingular conic discussed in the previous section. For arbitrary n we do a similar construction as is explained below. n Consider the linear system of quadrics in P containing a ﬁxed smooth quadric n n+1 Q0 of dimension n − 2. It maps P to a quadric Q in P . We may choose coordinates such that X 2 Q0 = V (z0) ∩ V ( zi ). i=1 so that the hyperplane H = V (z0) is the linear span of Q0. Then the linear system P 2 is spanned by the quadrics V ( zi ),V (z0zi), i = 0, . . . , n. It maps the blow-up n n+1 Pn 2 of P along Q0 to the quadric Q in P with equation t0tn+1 − i=1 ti = 0. The n rational map g : P 99K Q deﬁned by a choice of a basis of the linear system, can be given by the formula n X 2 2 [x0, . . . , xn] 7→ [ xi , x0x1, . . . , x0xn, x0]. i=1 Observe that the image of H is equal to the point a = [1, 0,..., 0]. The inverse of g is the projection map

n pra : Q 99K P , [z0, . . . , zn+1] → [z0, . . . , zn] n+1 from the point a. It blows down the hyperplane V (zn+1) ⊂ P to the quadric n Q0. Now consider the projection map prb : Q 99K P from a point b 6= a not lying

53 54 LECTURE 3. FIRST EXAMPLES in the hyperplane V (tn+1). Note that this hyperplane is equal to the embedded −1 tangent hyperplane TaQ of Q at the point a. The composition f = prb ◦ pra of the two rational maps is a quadratic transformation deﬁned by the homaloidal linear system of quadrics with the base locus equal to the union of Q0 and the point pra(b). For example, if we choose b = [0,..., 0, 1] so that pra(b) = [1, 0,..., 0], n n then the Cremona transformation f : P 99K P can be given by the formula n X 2 [x0, . . . , xn] 7→ [ xi , x0x1, . . . , x0xn]. (3.1) i=1

−1 −1 Note that f = pra ◦ prb must be given by similar quadratic polynomials. So the degree of f −1 is equal to 2. This is the reason for the name quadro-quadratic transformation. For example, if n = 2, if we rewrite the equation of Q0 in the form z0 = z1z2 = 0 and obtain the formula (1.7) for the standard quadratic transformation. Let us compute the multi-degree. For any general linear subspace L of codi- n mension k > 0, its pre-image under the proejction pb : Q 99K P is the intersection of Q with the subspace L0 = hL, bi. It is a quadric in this subspace. Since the point a does not belong to L0, the projection of this quadric from the point a is a quadric 0 in the projection of L from the same point. Thus dk = 2. This shows that the multi-degree of the transformation is equal to (2,..., 2). Let us conﬁrm it by the “high-tech” computations using the Segre classes. We have the exact sequence of normal sheaves

n n 0 → NQ0/H → NQ0/P → NH/P |Q0 → 0. This gives

n −1 −1 n n s(Q0, P ) = c(NQ0/P ) = c(NQ0/H ) c(NH/P ) ∩ [Q0]

−1 −1 X iX i X k+1 k k = (1+2h0) (1+h0) = (−2h0) (−h0) = (2 −1)(−1) h0. i≥0 i≥0 k≥0 where h0 is the class of a hyperplane section of Q0 and 1 stands for [Q0]. Note that n k under the homomorphism i∗ : A∗(Q0) → A∗(P ), the image of h0 ∈ An−3(Q0) is equal to 2hk+2. Thus we obtain, for k < n,

k k X k−ik n n−i dk = 2 − 2 i s(Bs(HX ), P )n−ih i=1

k k X k−ik n n n−i = 2 − 2 i [s(Q0, P )n−i + s(x0, P )n−i]h i=1 3.1. QUADRO-QUADRIC TRANSFORMATIONS 55

k k X k−ik i−2 i−1 n−k = 2 − 2 i (−1) 2(2 − 1) + [x0]h i=1 k k k k X ik X k−i ik k k k = 2 − 2 ( (−1) i ) + 2 2 (−1) i = 2 + 2 + 2(1 − 2 ) = 2. i=1 i=1 Let us consider some degenerations of the transformation given by (3.1). Let us n+1 take two nonsingular points a, b on an arbitrary irreducible quadric Q ⊂ P . We assume that b does not lie in the intersection of Q with the embedded tangent space −1 TaQ of Q at a. Let f = pra ◦ prb . The projection pa blows down the intersection TaQ ∩ Q to a quadric Q0 in the hyperplane H = pra(TaQ). If r = rank Q (i.e. n+1−r is the dimension of the singular locus of Q), then rank Q∩TaQ = r−1. Its singular locus is spanned by the singular locus of Q and the point a. The projection Q0 of Q ∩ TaQ is a quadric with singular locus of dimension n + 1 − r, thus, it is a quadric of rank equal to n − 1 − (n + 1 − r) = r − 2 in H. The inverse −1 n n transformation pra : P 99K Q is given by the linear system of quadrics in P which pass through Q0. So, taking a = [1, 0,..., 0] and b = [0,..., 0, 1] as in the non-degenerate case, we obtain that f is given by r−2 X 2 f :[x0, . . . , xn] 7→ [ xi , x0x1, . . . , x0xn]. (3.2) i=1 Note the special cases. If n = 2, and Q is an irreducible quadric cone, then r = 3 and we get the formula for the ﬁrst degenerate standard quadratic transformation (1.8). To get the second degenerate standard quadratic transformation, we should abandon the condition that b 6∈ TaQ. We leave the details to the reader. Example 3.1.1. Consider the Cremona transformation given by the quadric polynomials 2 (t3 − t1t2, t0(t1 − t2), t0(t2 − t3), t0(λt3 − t0). When λ 6= 0, the base scheme is reduced and is equal to the union of the smooth 2 conic C : t3 − t1t2 = t0 = 0 and the point [λ, 1, 1, 1]. When λ = 0, u 6= 0, the reduced base scheme is equal to C, however, it is not reduced. It contains an embedded point [0, 1, 1, 1] on C. It corresponds to the point b lying on TaQ. Example 3.1.2. Another example of a degenerate quadratic transformation is obtained by the homaloidal linear system of quadrics through a double structure Z n of codimension 2 linear subspace L of P . We consider only the case n = 3 and leave the computations in other cases to the reader. Consider the double structure on a line L deﬁned by the invertible sheaf OL(−1). Consider the exact sequences, obtained from exact sequence (2.28) after twisting by OP3 (1) and OP3 (2),

0 → JZ (2) → JL(2) → OL(1) → 0, 56 LECTURE 3. FIRST EXAMPLES

0 → JZ (1) → JL(1) → OL → 0. 0 3 0 3 It is easy to see that the homomorphism H (P , JL(i)) → H (P , OL(i − 1)) is surjective. Indeed, it assigns to a section s deﬁning the plane (quadric) containing L the section s¯ obtained by dividing s by the equation of the line and restricting 0 0 s¯ to L. This gives h (JZ (1)) = 1, h (JZ (2)) = 5. Using computations from Example 2.7.2 we obtain that the linear system of quadrics through Z and a point outside the plane corresponding to a non-zero section of JZ (1) is homaloidal. The corresponding Cremona transformation is the composition of the transformation

2 [x0, x1, x2, x3] 7→ [x1, x0x1, x0x2, x0x3] and a projective transformation. The equation of the line L here is x0 = x1 = 0, and the equation of the plane containing the double structure is x0 = 0. The 4 transformation is the composition of a rational map to a quadric of rank 3 in P , and the projection from a nonsingular point of the quadric.

It is not true that the multi-degree of a quadro-quadratic transformation in n P , n > 3 is always of the form (2,..., 2). For example, if n = 4, applying Cre- mona’s inequalities, we obtain d2 ≤ 4. A transformation of multi-degree (2, 3, 2) can be obtained by taking the homaloidal linear system of quadrics with the base scheme equal to a plane and two lines intersecting the plane at one point.

3.2 Quadro-quartic transformations

Suppose we have a Cremona transformation with multi-degree (2, m). An Φ-curve, the proper transform of a line in the target space, is of degree m. It is contained in the intersection of two quadrics, the proper transforms of planes. This shows that m ≤ 4. This can be also deduced from the log-concavity of the multi-degree. If m = 4, the formula for d2 in Proposition 2.3.1 shows that the reduced base scheme consists of isolated points. The image of a general plane under the transformation is given by a 3-dimensional linear system of conics without base points. Thus the 3 image must be a quartic surface, a projection of a Veronese surface to P . We know that there are no smooth homaloidal systems with isolated base points. Thus one of the base conditions must be that the proper transform of the homaloidal linear system to the blow-up of four points has base points on one of the exceptional locus. Since the intersection of two general quadrics from the linear system is a curve of arithmetic genus 1, and, on the hand it must be an irreducible rational curve (the pre-image of a line under the Cremona transformation), we see that the base condition must be that all quadrics are tangent at one point. Counting 3.2. QUADRO-QUARTIC TRANSFORMATIONS 57 dimension, we see that there must be four base points p1, . . . , p4 with a tangency condition at p4. If we assume that neither of the ﬁrst three points is inﬁnitely near, after composing the transformation with a projective transformation, we can give it by a formula

[x0, x1, x2, x3] 7→ [x0l(x1, x2, x3), x2x3, x1x3, x1x2], (3.3) where l is a linear form that vanishes at p4 = [1, 0, 0, 0] but does not not vanish at any of the first base points [1, 0, 0], [0, 1, 0], [0, 0, 1]. All quadrics in the linear system have the same tangent plane at p4 = [1, 0, 0, 0] equal to V (l). The conditions on l imply that l = at1 + bt2 + ct3, a, b, c 6= 0. We recognize in formula (3.3) a formula for a dilated quadratic transformation. Let us find the P -locus of the dilated quadratic transformation. Consider the planes spanned by two of the first three points pi, pj and p4 = [1, 0, 0, 0]. The restriction of the transformation to this plane is given by the linear system of conics through the points pi, pj, p4 with a fixed tangent direction at p4. It is a one- dimensional linear system which maps this plane to a line ìj in the target space. Also consider the restriction of the plane to the plane V (l). All quadrics are tangent to this plane at p4, so they restrict to conics with a singular point at p4. These conics consist of two lines passing through p4 and map the plane to a conic C in the target space. Thus the P -locus consists of four planes. Computing the jacobian of the transformation, we see that the expected degree of the P -locus is equal to 4, so there is nothing else in the P -locus. The reduced base locus of the inverse transformation T −1 must contain the image of the P -locus under T . We see that it consists of the union of three lines intersecting at one point (the image of the intersection point of the three planes hpi, pj, p4i). The image of the conic intersects all these lines. Let us see this in formulas. In homogeneous coordinates, our transformation is given by the polynomials (t0(at1 + bt2 + ct3), t2t3, t1t3, t2t3). Dividing by the first coordinate, and setting zi = ti/t0, we obtain a formula for the transformation in inhomogeneous coordinates

z2z3 z1z3 z1z2 (z1, z2, z3) 7→ (u1, u2, u3) = ( , , ). az1 + bz2 + cz3 az1 + bz2 + cz3 az1 + bz2 + cz3 It is easy to invert it. We ﬁnd

z1z2z3 au2u3 + bu1u3 + cu1u2 = . az1 + bz2 + cz3 This gives

au2u3 + bu1u3 + cu1u2 au2u3 + bu1u3 + cu1u2 au2u3 + bu1u3 + cu1u2 (z1, z2, z3) = ( , , ). u1 u2 u3 58 LECTURE 3. FIRST EXAMPLES

Homogenizing again, we ﬁnd the inverse is given by the polynomials

(t0t1t2t3, (at2t3+bt1t3+ct1t2)t1t2, (at2t3+bt1t3+ct1t2)t1t3, (at2t3+bt1t3+ct1t2)t2t3). We see that the transformation is dilated from the standard Cremona transformation. In notation of section 2.8, we have F0 = t0t1t2t3,Q = at2t3 + bt1t3 + ct1t2. The formula exhibits the base locus equal to the union of the conic V (at2t3 + bt1t3 + ct1t2, t0) and the three lines V (ti, tj), i, j 6= 0). The lines intersect at the point (1, 0, 0, 0). The conic intersects the three lines. The inverse map is given by the homaloidal linear system of quartics passing through the conic and passing with multiplicity 2 through the three lines. These surfaces are known to be Steiner quartic surfaces. They are projections of a Veronese 3 surface to P .

3.3 Quadro-cubic transformations

Now lets us consider some examples of transformations of type (2, 3), quadro- cubic transformations. The formula for d2 in Proposition 2.3.1 shows that the one- dimensional part of the base locus is a line. Counting the dimension, we see that the rest of the base locus must consist of three simple points, maybe infinitely near. We have seen such an example in Example 2.5.2. The transformation is given by a smooth homaloidal linear system with base scheme equal to the union of a line ` and three isolated points p1, p2, p3. The restriction of the linear system to a general plane is a linear system of conics passing through a fixed point. It maps the plane 3 onto a cubic scroll in P . This confirms that the inverse transformation is of degree 3. The P -locus consists of the union of four planes, the spans h`, pii and the span hp1, p2, p3i. The moving part of the restriction of the linear system to h`, pii is the linear system of lines through the point pi. The restriction of the linear system to the fourth plane is the linear system of conics through 4 points (the fourth point p4 is the intersection of ` with the forth plane). The planes in the P -locus are blown down to four lines in the target space. The first three lines are skew, and the fourth line intersects the first three lines at the points equal to the images of the lines hp4, pii, i = 1, 2, 3. The four lines form the base locus of the inverse transformation T −1. Using (2.26) we see that the homaloidal system of cubics defining the inverse transformation is not smooth. Formula (2.39) from Example 2.7.2 shows that the linear system of cubics through the forth line with double structure defined by a line bundle L of degree 3. This gives 7 conditions for containing the double structure and 9 more conditions to contain the the three lines. All of this can be checked by formulas. Without loss of generality, we may assume that the base locus of T consists of the line t0 − t1 = t1 − t2 = 0 and 3.4. BILINEAR CREMONA TRANSFORMATIONS 59 the three points [1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0]. The homaloidal linear system is generated by quadrics

Q : at0(t1 − t2) + b(t1 − t2)t3 + c(t0 − t1)t2 + (t0 − t1)t3 = 0.

A Cremona transformation T is given by choosing a basis in the space of such quadrics. For example, we can take

T :[x0, x1, x2, x3] 7→ [x0(x1 − x2), (x1 − x2)x3, (x0 − x1)x2, (x0 − x1)x3].

Dividing by the ﬁrst coordinate and using afﬁne coordinates z1 = t1/t0, z2, t2/t0, z3 = t3/t0, the transformation is given by the formula

(z1, z2, z3) 7→ (u1, u2, u3) = (z3, (1 − z1)z2/(z1 − z2), (1 − z1)z3/(z1 − z2)).

Its inverse is given by the formula

(z1, z2, z3) = (u1 + u1u2)/(u1 + u3), u1u2/u3, u1).

In homogeneous coordinates, we get the formula

[t0, t1, t2, t3] 7→ [t0t3(t1 + t3), t1(t0 + t2)t3, t1t2(t1 + t3), t0t1(t1 + t3)]

We see that the degree of this transformation is equal to 3. Its base locus consists of four lines

t0 = t1 = 0, t0 = t2 = 0, t1 + −t3 = t0 + t2 = 0, t1 = t3 = 0.

The ﬁrst three lines are skew. They all intersect the fourth line. All cubics from the homaloidal linear system are tangent along the last line. This deﬁnes a double structure on this line.

3.4 Bilinear Cremona transformations

These transformations generalize to any dimension cubic-cubic transformations in 3 P , one of which is the standard Cremona transformation Tst given by polynomials

(t1 ··· tn, t0t2 ··· tn, . . . , t0 ··· tn−1). as well as the transformation with base scheme equal to a double structure on a twisted cubic considered in Example 2.7.2. All of them belong to the same irreducible family of transformations with base locus in the Hilbert scheme of purely one-dimensional connected schemes Z with Hilbert polynomial P (t) = 6t − 2 60 LECTURE 3. FIRST EXAMPLES

0 and the additional condition that H (Z, OZ (2)) = 0. In particular, the degree of Z is equal to 6 and the arithmetic genus equal to 3. The last condition requires that Z does not lie on a quadric. These subschemes are examples of arithmetical Cohen-Macaulay schemes (ACM-schemes, for short). n Let Z be a closed subscheme of P . Recall that the homogenous ideal of Z is the graded ideal ∞ M 0 n IZ = Γ∗(JZ ) = H (P , JZ (k)) k=0 in the ring

∞ M 0 n ∼ Γ∗(OPn ) = H (P , OPn (k)) = S := C[t0, . . . , tn]. k=0

The graded quotient ring A(Z) = S/IZ is called the homogeneous coordinate ring of Z.

n Deﬁnition 3.4.1. A closed subscheme Z of P of pure dimension r is called arithmetically Cohen-Macaulay (ACM for short) if the homogeneous coordinate ring AZ is Cohen-Macaulay ring. This is equivalent to the following conditions:

(i) the canonical restriction map

0 n 0 n 0 A(Z)k = H (P , OPn (k))/H (P , JZ (k)) → H (Z, OZ (k))

is bijective.

i (ii) H (Z, OZ (j)) = 0 for 1 ≤ i ≤ r − 1 and j ∈ Z. Assume that codimZ = 2. It follows from the standard facts in commutative algebra that the projective dimension of the ring A(Z) is equal to 2. This means that IZ admits a resolution of length 2 of projective graded S-modules. Since each such module is free and it is isomorphic to the direct sum of graded S-modules 1 S[ai] Since JZ = Γ˜∗(JZ ) is the associated sheaf of the module IZ , we obtain a locally free resolution

m m+1 M M 0 → OPn (−ai) → OPn (−bj) → JZ → 0. (3.4) i=1 j=1 for some sequences of integers (ai) and (bj).

1 Recall that S[a] = S with shifted grading S[a]k = Sa+k. 3.4. BILINEAR CREMONA TRANSFORMATIONS 61

It is easy to see that the existence of such a resolution implies that Z is an ACM subscheme of pure codimension 2. The numbers (ai) and (bj) are determined from the Hilbert polynomials of Z. We will consider a special case of resolution of the form

n n+1 0 → OPn (−n − 1) → OPn (−n) → JZ → 0. (3.5)

It implies that

χ(OZ (k)) = χ(OPn (k)) − χ(JZ (k)) =

= χ(OPn (k)) + (n + 1)χ(OPn (k − n)) − nχ(OPn (k − n − 1)) n+k k k−1 = n − (n + 1) n + n n .

When n = 3, we get χ(OZ (k)) = 6k − 2, as we expect. Twisting (3.6) by OPn (n), we get an exact sequence

n n+1 0 → O n (−1) → O → J (n) → 0. (3.6) P Pn Z Taking cohomology, we obtain canonical isomorphisms

0 n ∼ 0 n n+1 ∼ n+1 H ( , J (n)) = H ( , O n ) = , P Z P P C n−1 n ∼ n n n ∼ n H (P , JZ ) = H (P , OPn (−1 − n)) ) = C . ∼ The latter isomorphism follows from Serre’s Duality since OPn (−1 − n) = ωPn . Let

0 n V = H (P , JZ (n)), n−1 n W = H (P , JZ ).

Then we can rewrite (3.6) in the form

0 → OPn (−1) ⊗ W → OPn ⊗ V → JZ (n) → 0. (3.7)

n ∨ Let P = P(E ) = |E|. Passing to the maps of stalks, we see that the exact sequence is deﬁned by a linear map

W ⊗ E → V. or, equivalently by a tensor

a ∈ W ∨ ⊗ E∨ ⊗ V. 62 LECTURE 3. FIRST EXAMPLES

We can also view it as a linear map

a : E → Hom(W, V ) = W ∨ ⊗ V. (3.8)

Choosing coordinates t0, . . . , tn in E, a basis in W and a basis in V , the map a is given by a matrix A = (aij(t)) of size (n + 1) × n with entries linear functions aij(t). For any point x = [x0, . . . , xn], we denote by A(x) the matrix (aij(x)). Tensoring exact sequence (3.7) by the residue ﬁeld of a closed point, we ﬁnd that

n {x ∈ P : rankA(x) < n} = Supp(Z). We can do better. Recall that exact sequence (3.6) comes from a projective resolution M n A n+1 0 → S[−n − 1] → S[−n] → IZ → 0, where the map A is given by our matrix A(t) of linear forms. By Hilbert-Birch Theorem (see [Eisenbud, Com. Algebra,Theorem 20.15]), this implies that IZ is generated by maximal minors Di of the matrix A. Since the ideal IZ is saturated, it determines uniquely the scheme Z (see [Hartshorne], Chap. 2, Exercise 5.10]. Thus Z is equal to the base scheme of a rational map

∨ Ta : |E| → |V | deﬁned by the n-dimensional linear system |V | = |JZ (n)| generated by the maximal minors of A. Explicitly, the map Ta is deﬁned by

∨ t Ta([x]) = |Coker(a(x))| ∈ |V | = Ker( a(x)), where ta(x): V ∨ → W is the transpose of a(x). n Remark 3.4.1. The Hilbert scheme of ACM subschemes Z of P admitting a resolution (3.7) is isomorphic to an open subset of the projective space of (n + 1) × n of matrices A(t) of linear forms such that the rank of A(t) is equal to n for an n open non-empty subset of P . Modulo the action by GL(n + 1) × GL(n) by left and right multiplication. It is a connected smooth variety of dimension n(n2 − 1) (see [Peskine-Szpiro], Inventiones, 1974], or [Ellingsrud, Ann. Ec. Norm. Sup. 8 (1975)]).

Proposition 3.4.1. The map Ta is a birational map. The multi-degree is equal to n (dk) = ( k ). Proof. Let us view the tensor a as as a bilinear map

E ⊗ V ∨ → W ∨. (3.9) 3.4. BILINEAR CREMONA TRANSFORMATIONS 63

If we ﬁx a basis in E, a basis in V ∨, and a basis in W , then the bilinear map is deﬁned by n square matrices B1,...,Bn of size n + 1. If we write A(t) as t0A0 + ... + tnAn, and write Aj in terms of columns Aj = [Aj1,...,Ajn], then

Bi = [A0i,...,Ani], i = 1, . . . , n.

The linear map a can be considered as a bilinear map (or, as a set of n bilinear forms)

n+1 n+1 n C × C → C , (~x,~y) 7→ (~x · B1 · ~y, . . . , ~x · Bn · ~y).

The map Ta is given by assigning to [~x] ∈ |E| the intersection of the kernels of the matrices ~x · Bi. The common kernel is one-dimensional if [~x 6∈ Supp(Z). This is equivalent to that the n vectors ~x · Bi are linearly independent. The inverse map is deﬁned by assigning to [~y], the intersection of the kernels of the matrices t Bi · ~y = ~y · Bi. Since the rank of a matrix and its transpose are the same, we obtain, that the inverse map is well-deﬁned on the complement of Supp(Z). −1 Note that Ta = Ta if we can choose coordinates in V,E,W such that the matrices Bi are symmetric. n n ∨ The graph of the map Ta is the closed subset of P × P = |E| × |V | given by n divisors of type (1, 1) corresponding to the matrices Bi considered as bilinear ∨ forms on E × V . The matrices B1,...,Bn are linearly independent since

n n n X X X λiBi = [ λiA0i,..., λiAni] = 0 i=1 i=1 i=1 for some ~λ = (λ1, . . . , λn) 6= 0 implies that A0 · ~λ = ... = A0 · ~λ = 0, hence A(x) · ~λ = (x0A0 + ... + xnAn) · ~λ = 0. This means that rankA(t) < n for n all [x] ∈ P contradicting our assumption. Now we can compute the cohomology class of the graph. It is equal to

n n X n i n−i (h1 + h2) = k h1h2 . k=0

Consider the tensor a as a linear map

ψ : W → E ⊗ V ∨ = (E∨ ⊗ V )∨. (3.10)

We view the target space as the space of bilinear forms on E∨ × V , or as a space of linear maps E∨ → V ∨. By choosing bases in E and V , it can be identiﬁed 64 LECTURE 3. FIRST EXAMPLES with the linear space of square matrices of size n + 1. The injectivity of the map E → Hom(W, V ) implies that ψ is injective, so we can identify W with its image. It is a linear n-dimensional subspace of the space of bilinear forms. Let Dk ⊂ W be the closed subvariety of bilinear forms of rank ≤ k. Its equations in the afﬁne ∼ n+1 space W = C are k × k-minors of the matrix representing the bilinear form. Let Dk be the image of Dk in the projective space |W |. We have a regular map

∨ l : Dn \ Dn−1 → |E| (resp. r : Dn \ Dn−1 → |V |) which assigns to z = [w] the left (resp. the right) kernel of the bilinear form ψ(w). By deﬁnition, the image of the map l (resp. r) is contained in the base locus of the −1 rational map Ta (resp. Ta ). ∨ −1 Note the special case when E = V and Ta = Ta . In this case W lies in the space of symmetric bilinear forms and the two maps l and r coincide. n Example 3.4.1. Consider the standard Cremona transformation of degree n in P given by x0 ··· xn x0 ··· xn Tst :[x0, . . . , xn] 7→ [ ,..., ]. (3.11) x0 xn

In afﬁne coordinates, zi = ti/t0, it is given by the formula

−1 −1 (z1, . . . , zn) 7→ (z1 , . . . , zn ).

The transformation Tst is an analogue of the standard quadratic transformation of the plane in higher-dimension. The base ideal of Tst is generated by the polynomials t1 ··· tn, . . . , t0 ··· tn−1. It is equal to the ideal generated by the maximal minors of the n × n matrix   t0 0 ··· 0  0 t1 ... 0     . . .. .  A(t) =  . . . .     0 0 . . . tn−1 −tn −tn ... −tn

The matrix A(t) deﬁnes a resolution (3.7) of the base scheme of Tst equal to the union of the coordinate subspaces of codimension 2. It follows from the proof of Proposition 3.4.1 that the graph of Tst is isomorphic n n to the closed subvariety X of P × P given by n bilinear equations

xiyi − xnyn = 0, i = 0, . . . , n − 1.

n n It is a smooth subvariety of P × P isomorphic to the blow-up of the union of ∗ n+1 n coordinate subspaces of codimension 2. The action of the torus (C ) on P 3.4. BILINEAR CREMONA TRANSFORMATIONS 65

(by scaling the coordinates) extends to a biregular action on X. The corresponding toric variety is a special case of a toric variety deﬁned by a fan formed by fundamental chambers of a root system of a semi-simple Lie algebra. In our case the root system is of type An, and the variety X is denoted by X(An). In the case n = 2, the toric surface X(A2) is a Del Pezzo surface of degree 6 isomorphic to the blow-up of 3 points in the plane, no three of which are collinear. It is classically known and, it is an easy fact, that a Del Pezzo surface of degree 8 6 embeds by the anti-canonical linear system into P as a complete intersection of 2 2 the Segre variety s(P × P ) and two hyperplanes. The previous proof gives the n2+2n analogous fact for any X(An). Namely, X(An) embeds in P as a complete n n intersection of the Segre variety s(P × P ) and a linear space of codimension n. By computing the canonical class of X(An), one can show that the embedding is given by the anti-canonical linear system. 3 Example 3.4.2. Let Z be a connected closed one-dimensional subscheme of P 0 2 with Hilbert function 6t − 2 and h (OZ ) = 1. In particular, [Z] = 6h and the 1 arithmetic genus h (OZ ) = 3. Assume Z is arithmetically Cohen-Macaulay. Then the canonical map 0 3 0 3 φj : H (P , OP3 (j)) → H (P , OC (j)) is surjective for all j ∈ Z. Since deg OC (j) = 6j > 2pa(Z) − 2 = 4, we have 1 0 h (OC (j)) = 0 . Thus h (OC (j)) = χ(OC (j)) = 6j − 2. Taking j = 1, this implies that Z is linearly normal, i.e. it is not a projection of any scheme from a 0 higher-dimensional space. Taking j = 2, this implies that h (JZ (2)) = 0, i.e. Z does not lie on a quadric. When Z is reduced and irreducible, these two conditions imply that Z is ACM (see [Ellingsrud, Ann. Ec. Norm. Sup., 1974], p. 430). Taking the cohomology of the exact sequence

0 → JZ → OPn → OZ → 0, we obtain 2 1 h (JZ ) = h (OZ ) = 3. Using resolution (3.6), we obtain 3 3 2 X 3 X 0 h (JZ ) = h (OP3 (−bj) = h (OP3 (bj − 4) = 3. j=1 j=1

The only solution is b1 = b2 = b3 = 4. Applying Riemann-Roch, we see that 0 h (JZ (3)) = 4, hence 4 3 4 X 0 X 0 X 0 4 = h (OP3 (3 − ai)) − h (OP3 (3 − bj)) = h (OP3 (3 − ai)). i=1 j=1 i=1 66 LECTURE 3. FIRST EXAMPLES

P4 0 Since Z does not lie on a quadric, 0 = i=1 h (OP3 (2 − ai)), hence all ai > 2. Together, this easily implies that a1 = ... = a4 = 3. Thus the resolution (3.4) equals the resolutiion (3.7). So the linear system of cubics through R is a homoloidal linear system and the multi-degree of the Cremona transformation is equal to (1, 3, 3, 1). Because the degree of the transformation and its inverse are equal to 3, such transformation are classically known as cubo-cubic transformations. Assume Z is a smooth curve C and let us describe the P -locus of the corresponding Cremona transformation. Obviously, any line intersecting C at three distinct points (a trisecant line) must be blown down to a point (otherwise a general cubic in the linear system intersects the line at more than 3 points). Consider 3 the trisecant locus Tri(C) of C, the closure in P of the union of lines intersecting C at three points. Note that no line intersects C at > 3 points because the ideal of C is generated by cubic surfaces. Consider the linear system of cubics through C. If all of them are singular, by Bertini’s Theorem, there will be a common singular point at the base locus, i.e. at C. But this easily implies that C is singular, contradicting our assumption. Choose a nonsingular cubic surface S containing C. By 2 adjunction formula, we have C = −KS ·C +deg KZ = 6+4 = 10. Take another cubic S0 containing C. The intersection S ∩ S0 is a curve of degree 9, the residual curve A is of degree 3 and C +A ∼ −3KS easily gives C ·A = 18−10 = 8. Note that the curves A are the proper transforms of lines under the Cremona transformation. So they are rational curves of degree 3. We know that the base scheme of the inverse transformation f −1 is a curve of degree 6 isomorphic to C. Replacing f with f −1 we know that the image of a general line ` under f is a rational curve of degree 3 intersecting C0 at 8-points. These points are are the images of 8 trisecants intersecting `. This implies that the degree of the trisecant surface Tri(C) is equal to 8. Since the degree of the determinant of the Jacobian matrix of a transformation of degree 3 is equal to 8, we see that there is nothing else in the P -locus. The linear system of planes containing ` cuts out on C a linear series of degree 6 with moving part of degree 3. It is easy to see, by using Riemann-Roch, that 1 any g3 on a curve of genus 3 must be of the form |KC − x| for a unique point x ∈ C. Conversely, for any point x ∈ C, the linear system |OC (1) − KC + x| is of dimension 0 and of degree 3 (here we use that |OC (1)| = |KC + a|, where a is not effective divisor class of degree 2). Thus it deﬁnes a trisecant line (maybe tangent at some point). This shows that the curve R parameterizing trisecant lines is isomorphic to C. This agrees with the fact that R must be isomorphic to the base curve of the inverse transformation. The Cremona transformation can be resolved by blowing up the curve C and then blowing down the proper transform of the surface Tri(C). The exceptional divisor is isomorphic to the minimal ruled surface with the base C. It is the universal family of lines parameterized by C. Its image 3.4. BILINEAR CREMONA TRANSFORMATIONS 67

3 0 0 in the target P is surface Tri(C ), where C is the base locus of of the inverse 0 transformation (the same curve only re-embedded by the linear system |KC + a |, 0 where a ∈ |KC − a|). 0 3 ∼ 3 Let W = H (P , JC ) = C . The hypersurface D2 is a plane quartic curve. 3 3 The images of maps l : |W2 → P and r : |W2 → P are isomorphic to C. The 3 two embeddings of D2 in P are given by |KC + a| and |2KC − a|. This gives an equation of a plane quartic curve as the determinant of a square matrix of linear forms. Up to a natural equivalence of such representations (replacing the matrix of linear forms by row and column transformations) the set of such representations is parameterized by Pic2(C) \ Θ, where Θ is the hypersurface of effective divisor 0 classes of degree 2). If a = , then 2a = KC , so that a is a theta characteristic. Since a is not effective, it is an even theta characteristic. It is known that the number of even theta characteristics on a nonsingular curve of genus 3 is equal to 36. In ∨ 0 ∨ ∨ this case we can identify the spaces |KC − a| = |U| and |KC − a | = |V |, and the matrix defining D3 can be chosen to be a symmetric matrix. 3 Example 3.4.3. Let Z be the union of 4 skew lines in P and their two transversals, i.e. lines intersecting all the four lines. It is easy to compute that Z is a curve of arithmetic genus 3, obviously of degree 6. Assume that the four lines do not lie on a quadric. Then Z is arithmetically Cohen-Macualy and can be taken as the base scheme of a cubo-cubic Cremona transformation. The construction of two transversals to 4 skew lines `1, . . . , `4 is as follows. Choose a set of three lines among the given 4 lines, say `1, `2, `3. By counting constants, we see that there is a unique quadric Q containing these lines. Since the lines are skew, the quadric is nonsingular, and the three lines belong to one of its rulings by lines. The line `4 intersects the quadric Q at two points p, q. The two lines passing through p and q from the other ruling are the two transversals. When the line `4 is tangent to Q, we have p = q, so the two transversals degenerate to a double structure of one transversal. Obviously, any line on Q from the ruling containing the first three lines belongs to the P-locus. Thus Q belongs to a P-locus. Choosing four subsets of three lines among `1, . . . , `4, we obtain that the union of 4 quadrics belongs to P -locus. Since the degree of the determinant of the jacobian matrix is equal to 8, there is nothing else in the P-locus. 3 Example 3.4.4. Consider the double structure Z on a twisted cubic curve C in P ∼ defined by an invertible sheaf L = OP1 (−4). Exact sequence (2.28) gives an exact sequence

0 → OP1 (−4) → OZ → OC → 0. 0 0 Computing cohomology, we ﬁnd that, for n ≥ 1 h (OZ (n)) = h (OP1 (3n)) + 0 h (OP1 (3n − 4)) = 6n − 2. Thus the Hilbert polynomial of Z is the same as for 68 LECTURE 3. FIRST EXAMPLES the subscheme deﬁned by a resolution (3.7). Tensoring the exact sequence

0 → JZ → JC → OP1 (−4) → 0 0 0 0 by OP3 (3), we get h (OZ (3)) = h (OP1 (9)) − h (OP1 (5)) = 4. Here we used 1 that h (JZ (3)) = 0. This can be veriﬁed by using the exact sequences

0 → JZ (3) → JC (3) → OP3 (−1) → 0, 1 and the fact that h (JC (3)) = 0. It follows from calculations in Example 2.7.2 that the linear system |JZ (3)| is homaloidal. 3 ∨ Let us write P as |U| = P(U ) for some two-dimensional linear space U and identify C with the image of the Veronese map

1 3 3 3 v3 : P = |U| → P = |S U|, [u] 7→ [u ]. Binary cubic forms φ ∈ S3U with three distinct zeros in |U ∨| form an orbit with respect to SL(U). Each element of this orbit can be uniquely written as a sum of two forms u3 and v3, hence lies on a unique secant of C. A form φ with two distinct zeros form another orbit. The closure of this orbit is the tangential surface of C, the closure of the union of tangent lines to C. Finally, the curve C itself is the third orbit, the unique closed orbit. If we choose a basis η0, η1 in U and view an element of S3U as a binary form

3 2 2 3 φ = a0η0 + 3a1η0η1 + 3a2η0η1 + a3η1 (3.12) on the dual space U ∨, then the equation of the tangential scroll Tan(C) of C is given by the discriminant of the binary cubic form

2 2 3 3 2 2 D4 = 3t1t2 − 4t0t2 − 4t1t3 − t0t3 + 6t0t1t2t3 = 0. (3.13) For this reason surface Tan(C) is also called the discriminant quartic surface. The curve C is its cuspidal curve. Consider a natural bilinear map of SL(U)-modules

S3U ⊗ S3U → S2U (3.14) equal to the projection in the Clebsch-Gordan decomposition of the representations of SL(U): 3 3 6 4 2 S U ⊗ S U → S U ⊕ S U ⊕ S U ⊕ C.

In coordinates, it is given by the second transvectant t2. If we view an element of S3U as a binary form

3 2 2 3 φ = a0η0 + 3a1η0η1 + 3a2η0η1 + a3η1 (3.15) 3.4. BILINEAR CREMONA TRANSFORMATIONS 69

∨ on the dual space U with coordinates η0, η1, then (3.14) is given by the formula

(φ, ψ) 7→ φ00ψ11 − 2φ01ψ10 + φ11ψ00, where the subscripts indicate the second partial derivatives with respect to η0, η1. 2 2 Explicitly, (φ, ψ) is mapped to c0η0 + c1η0η1 + c2η1, where

c0 = a0b2 − 2a1b1 + a2b0

c1 = a0b3 − a1b2 − a2b1 + a3b0

c2 = a1b3 − 2a2b2 + a3b1.

The pairing defines our tensor a ∈ S2U ⊗(S3U)∨ ⊗(S3U)∨. In previous nota- tions W = S2U ∨,E = S3U, V = S3U ∨. Note that the SL(U)-representations U ∨ 2 ∼ and U are canonically isomorphic via the determinant map U ⊗ U → Λ U = C. The three 4 × 4-matrices Bi defining the three bilinear forms are the matrices of the bilinear forms c0, c1, c2. The matrix A of linear forms defining the resolution (3.7) is the following   t2 t3 0 −2t1 −t2 t3  A(t) =    t0 −t1 −2t2 0 t0 t1 Computing the maximal minors we find them to be proportional to the partial derivatives of the quartic form D4

3 2 2 2 2 2 3 2 3t1t2t3 − 2t2 − t0t3, t0t2t3 + t1t2 − 2t1t3, t0t1t3 + t1t2 − 2t0t2, 3t0t1t2 − 2t1 − t0t3.

The linear system formed by the partial derivatives is our homaloidal linear system ∼ ∨ |JZ (3)|. Since E = V , we can canonically identify the source space |E| with the 3 ∨ target space P = |V |. The columns of the matrix A(t) give us the relations between the partial deriva- 3 tives. As in the proof of Theorem 3.4.1, we ﬁnd that the blow-up BlZ P is isomor- 3 3 phic to the subscheme of P × P given by three bilinear equations

t3u0 − 2t1u1 + t0u2 = 0,

t0u3 − t1u2 − t2u1 + t3u0 = 0,

t1u3 − 2t2u2 + t3u1 = 0.

3 3 Its intersection with the diagonal of P ×P is isomorphic to the closed subscheme 3 Y of P given by the equations, obtained from the previous equations by setting ui = ti, i = 0, 1, 2, 3. It is easy to see that these equations define a twisted cubic. 3 3 By taking the affine pieces Uij : ui = tj = 1 in P × P , and computing the matrix of the partial derivatives at each point of Y , we find that that Y is a singular 70 LECTURE 3. FIRST EXAMPLES

3 curve on BlZ P . Note that the exceptional divisor is isomorphic to the minimal rational ruled surface F2. Its exceptional section is the curve of singularities of 3 BlZ P . The proper transform of Tan(C) under the first projection on the blow-up scheme is its resolutions of singularities. It is isomorphic to a quadric. Under the 3 second projection to P , it is blown-down to C. The restriction of the linear system |JZ (3)| to a general plane H is the linear system of cubic curves in H passing through the double structure on the set of three points H ∩ C. It consists of three points with given tangent directions. A Cremona transformation T defined by the homaloidal linear system |JZ (3)| maps H to a cubic surface with 3 double points projectively isomorphic to the surface w3 + xyz = 0. The P -locus of T is the discriminant surface Tan(C). It should be thought as a degeneration of the trisecant octic surface of a nonsingular curve of genus 3 and 3 3 degree 6 in P . It is easy to see that the normal bundle of C in P is isomorphic to OP1 (5) ⊕ OP1 (5). Using the computations in Example 2.7.2, we obtain that 3 the exceptional divisor E1 of the blow-up of X = BlC P is isomorphic to F0 = 1 1 P × P . The section R of E1 defined by the invertible sheaf OP1 (−4) defining the double structure is a curve on E1 of bi-degree (1, 1). The restriction of the 3 blowing down morphism π1 : X → P to E1 is defined by one of the projections 1 1 1 P × P → P . The proper transform F of Tan(C) on X is isomorphic to F0. The curve R on F0 is also of type (1, 1). The two surfaces E1 and F are tangent to each other along the curve R (recall that C is a cuspidal curve on Tan(C)). 0 The exceptional divisor E2 of the blow-up ν : X = BlRX → X is also 0 0 isomorphic to F0. The proper transform E1 of E1, the proper transform F of F intersect E2 along the same curve isomorphic to R. The proper transform of the 0 0 homaloidal linear system on X is equal to the linear system H = |3H −E1 −2E2|, 3 where H is the full inverse image of a plane in P . One checks that the restriction of 0 H to E1 is the linear system |3f|, where f is a fibre of one of the projections F0 → 1 0 P . The variety X defines a smooth resolution of the Cremona transformation. 0 3 0 3 The morphism π : X → P is the composition π1 ◦ν. The morphism σ : X → P 0 0 0 is the composition σ1 ◦ ν , where ν : X → X is the blow-down of the proper 0 3 0 transform of F and σ1 : X → P is the blowing down of the image of E1 on X to 0 3 0 a twisted cubic C in the target P . The image of E2 is the tangent surface of C .

ν ν0 0 ) ~ T 3 X / X BlZ P = ΓT

π1 σ1 3 T 3 y P s / P 3.5. MONOMIAL BIRATIONAL MAPS 71

3 0 0 Note that the blow-up BlZ P is obtained from X by blowing down E1 to the curve 3 3 of singularities of BlZ P . It resolves the singularities of BlZ P . One should think 0 0 about E1 as an analog of a (−2)-curve, and F as analog of (−1)-curve. Finally observe that the determinantal quartic hypersurface D3 is equal to the 3 double conic. The 2-dimensional linear system of quadrics on P is equal to the net |JR(2)|. We may think that the double structure on R is the scheme of singular points of quadrics in the net. The situation is very similar to the resolution of the planar quadratic map with one inﬁnitely near point. Example 3.4.5. Let n = 4. The Hilbert function of Z is equal to 5(k − 1)2. Thus it is a surface of degree 10 and χ(OZ ) = 5. It admits a degeneration into the union of 10 coordinate planes. The base ideal is generated by ﬁve polynomials of degree 4, the maximal minors of a matrix A(t) of size 5×4. If Z is reduced, it is birationally 3 isomorphic to the discriminant surface D4 ⊂ P of degree 5. It is known that, for a general matrix A(t), the discriminant surface is smooth.

3.5 Monomial birational maps

∼ n Let XΣ be a toric variety defined by a fan Σ in a lattice N = Z . Let g ∈ GLn(N) be an automorphism of N. It defines an isomorphism of the toric varieties XΣ 0 and XΣ0 , where Σ = g(Σ). Let us identify these two toric varieties by means of this isomorphism. Let Π be the minimal common subdivision of Σ and Σ0. Then projections π : XΠ → XΣ and σ : XΠ → XΣ0 define a smooth resolution of the monomial Cremona transformation f = σ ◦ π−1. On the dense open torus orbit ∗ n identified with (C ) with coordinates z1, . . . , zn, the map f given by monomials whose exponent vectors are the rows of the matrix g

m1 mn f :[z1, . . . , zn] 7→ [z ,..., z ].

n Let us take XΣ to be equal to P with the toric structure deﬁned by the standard fan with the 1-skeleton consisting of rays R≥0e1,..., R≥0en, R≥0e0, where n (e1,..., en) is the standard basis of R and e0 = −(e1 + ··· + en). Recall that ample torus-linearized invertible sheaves on XΠ corresponds to convex lat- ∨ tice polytopes in the space MR, where M = N is the dual lattice. The sheaf ∗ π OPn (1) corresponds to the simplex Σn spanned by the vectors 0, e1,..., en and ∗ the sheaf σ (OPn (1) corresponds to the simplex g(Σn) spanned by the vectors k n−k 0, g(e1), . . . , g(en). The intersection h1h2 on XΠ is equal to the mixed volumes

dk = Vol(Σn, k; g(Σn), n − k). If g is the linear map v 7→ −v, we get the standard Cremona transformation of n degree n. From our computation, we see that dk = k . A formula for computing 72 LECTURE 3. FIRST EXAMPLES the multidegrees of a monomial Cremona transformation was given recently by P. Alufﬁ (arXiv:1308.4152). The formulas in Remark 1.4.1 allow us to compute the degree of a monomial map. The group GL(n, Z) is generated by the transformations

g1 :(m1, . . . , mn) 7→ (m2, m3, . . . , mn, m1),

g2 :(m1, . . . , mn) 7→ (m1, m2 + m3, m3, . . . , mn),

g3 :(m1, . . . , mn) 7→ (−m1, m2, m3, . . . , mn).

(see [Coxeter], 7.2). The corresponding monomial Cremona transformations are

f1 :(z1, . . . , zn) 7→ (z2, z3, . . . , zn, z1),

f2 :(z1, . . . , zn) 7→ (z1, z2z3, z3, z4, . . . , zn), −1 f3 :(z1, . . . , zn) 7→ (z1 , z2, z3, . . . , zn).

n The ﬁrst transformation is a projective automorphism of P , the other ones are quadratic transformations. Thus the subgroup of the group Cr(n) of Cremona n transformations of P (the Cremona group of degree n) generated by monomial transformations is generated by projective and quadratic transformations. Lecture 4

Involutions

4.1 De Jonquieres` involutions

We will be often using the following classical notion of harmonically conjugate pairs of points on the projective line. 1 1 1 Let γ : P → P be an involution of P . It is given by the deck transformation 1 1 1 of a degree two map u : P → P which, in its turn, is given by a linear series g2 and a choice of its basis. Clearly, it is uniquely defined by a choice of two members 1 a + b and c + d of g2. In particular, we can choose the two members to be the two divisors of the form 2a and 2c defined by the two ramification points of u. Another choice is to choose a reduced divisor a + b and one non-reduced divisor 2c. Then the second non-reduced divisor 2d is defined uniquely. This suggests the two ways 1 to define an involution on P . Choose two distinct points a, b which will be in involution, and assign to a point c the unique point γ(c) such that 2c and 2γ(c) are 1 the two non-reduced divisors of g2. Another way, choose two distinct points a, b 1 and define the unique g2 containing 2a and 2b. 1 Definition 4.1.1. Two pairs {a, b} and {c, d} points on P are called harmonically 1 1 conjugate, if there exists a g2 such that 2a, 2b, c + d ∈ g2. 2 2 Lemma 4.1.1. Let a, b be the zeros of a binary form αt0 +2βt0t1 +γt1 and c, d be 0 2 0 0 2 the zeros of a binary form α t0 + 2β t0t1 + γ t1. Then the pairs {a, b} and {c, d} are harmonically conjugate if and only if

αγ0 + α0γ − 2ββ0 = 0. (4.1)

2 2 Proof. Consider the Veronese map [t0, t1] 7→ [t0, t0t1, t2] with the image equal 2 to the conic C = V (x0x2 − x1). The points a, b are mapped to the intersection 1 points A, B of the line V (αx0 + 2βx1 + γx2) with the conic. The linear series g2

73 74 LECTURE 4. INVOLUTIONS containing 2a and 2b is the pre-image of the linear series on C cut out by the pencil of lines through the intersection point Q of the tangent lines of C at the points p1 and p2. This is the point [u0, u1, u2] such that its polar line with respect to C is equal to the line hp1, p2i. The equation of the polar line is u2x0−2u1x1+u0x2 = 0. 1 Thus [u0, u1, u2] = [α, −2β, γ]. A divisor c + d belongs to g2 if and only it is mapped to the intersection points C,D of the conic with a line passing through Q. This happens if and only if α0α − 2ββ0 + γγ0 = 0.

Q C AB

Note that, as a corollary, we obtain that the deﬁnition is symmetric with respect 1 to the pairs {a, b} and {c, d}. Also, if we choose the coordinates t0, t1 in P such that {a, b} = V (t0t1), then a member c + d of the pencil generated by 2a and 2b 2 2 are zeros of a binary form λt0 + µt2. In afﬁne coordinates a = 0, b = ∞, c = z, d = −z, and the cross-ratio

(c − a)(b − d) (z − 0)(∞ + z) R(c, a, b, d) = = = −1. (c − b)(a − d) (z − ∞)(0 + z)

This gives another equivalent definition of harmonically conjugate pairs. Another 1 equivalent definition is that the double cover of P ramified√ at the union of two pairs is an elliptic curve with complex multiplication by −1. n Let X be a reduced irreducible hypersurface of degree m in P which contains a point o of multiplicity m − 1. We call such a hypersurface submonoidal. For example, every smooth hypersurface of degree ≤ 3 is submonoidal. Let us choose the coordinates such that o = [1, 0,..., 0]. Then X is given by an equation

2 Fm = t0am−2(t1, . . . , tn) + 2t0am−1(t1, . . . , tn)

+am(t1, . . . , tn) = 0, (4.2) where the subscripts indicate the degrees of the homogeneous forms. For a general point x ∈ X consider the intersection of the linear span `x = ho, xi with X. It contains o with multiplicity m − 2 and the residual intersection is a set of two 4.1. DE JONQUIERES` INVOLUTIONS 75 points a, b in `x. Consider the involution on `x with fixed points a, b. Define T (x) to be the point on `x such that the pair x and T (x) are conjugate with respect to the involution. In other words, the pairs {a, b} and {x, T (x)} are harmonically conjugate. We call it a De Jonquieres` involution (observe that T = T −1). Let us find an explicit formula for the De Jonquieres` involution which we have defined. Let x = [x0, . . . , xn] and let [u + vx0, vx1, . . . , vxn] be the parametric equation of the line `x. Plugging in (4.1), we find 2 m−2 m−1 (u + vx0) v am−2(x1, . . . , xn) + 2(u + vx0)v am−1(x1, . . . , xn) m +v am(x1, . . . , xn) = 0. m−2 Canceling v , we see that the intersection points of the line `x with X are the two points corresponding to the zeros of the binary form Au2 + 2Buv + Cv2, where (A, B, C) = (am−2(x), x0am−2(x) + am−1(x),Fm(x)). The points x and T (x) corresponds to the parameters satisfying the quadratic equation A0u2+2B0uv+C0v2 = 0, where AA0+CC0−2BB0 = 0. Since x corresponds to the parameters [0, 1], we have C0 = 0. Thus T (x) corresponds to the parameters [u, v] = [−C,B], and

T (x) = [−C + Bx0, Bx1, . . . , Bxn]. Plugging the expressions for C and B, we obtain the following formula for the transformation T 0 x0 = −x0am−1(x1, . . . , xn) − am(x1, . . . , xn), 0 xi = xi(am−2(x1, . . . , xn)x0 + am−1(x1, . . . , xn)), i = 1, . . . , n.

n−1 Observe that T is a dilation of the identity transformation of P . The divisors from the homaloidal linear system are hypersurfaces of degree m which have singular points of multiplicity ≥ m − 1 at o. Such hypersurfaces are classically known as monoidal hypersurfaces. In the notation of section 2.8, we have Gi = ti, i = 1, . . . , n, F0 = −x0am−1(x1, . . . , xn) − am(x1, . . . , xn),Q = am−2(x1, . . . , xn)x0 + am−1(x1, . . . , xn). So, d = 1, r = m − 1. Since the identity transformation has no base points, we obtain that the multi-degree is equal to (m, m, . . . , m). In particular, we see that any such vector is realized as the multi-degree of a Cremona transformation. Assume that m > 2. The base scheme of the homaloidal linear system H deﬁning T is equal to

Bs(H) = V (t0am−1 + am, t1(am−2t0 + am−1), . . . , tn(am−2t0 + am−1)). 76 LECTURE 4. INVOLUTIONS

Note that V (a t + a ) = V ( ∂Fm ) is the polar hypersurface P (X) of X m−2 0 m−1 ∂t0 o with respect to the point x. Also, V (t a + a ) = V (−F + ∂Fm ). Thus 0 m−1 m m ∂t0 the base scheme contains the intersection X ∩ Po(X). Set-theoretically, this intersection is equal to the locus of points x ∈ X such that the embedded tangent hyperplane TxX contains the point o. In particular, all singular points of X are contained in the base locus. In other words, the intersection X ∩ Po(X) \{o} n−1 consists of the points where the projection map pro : X \{o} → P is not smooth. Consider the enveloping cone ECo(X) of X at the point o. It is the union of lines through o and a point on X ∩ Po(X). Each such line either contained in X, or intersects it only at one point a besides o. In the ﬁrst case, the line is contained in the base locus of T . In the second case, it is blown down to the point a. This easily follows from the formula for the transformation (use that AC = B2 and we 0 0 2 can take A = 2C,B = B). In our case ECo(X) = V (am−2 − am−1am). Its degree is equal to 2(m−1). The other part of the P -locus is the polar hypersurface V ( ∂Fm ). It is blown down to the point o. Its degree is equal to m − 1. There is ∂t0 nothing else in the P -locus. When n > 2, we have to take the polar hypersurface with multiplicity n − 1. Example 4.1.1. Assume that n = 2. Then the variety X is a hyperelliptic curve of genus g = m − 2 (a rational or elliptic curve if d = 2 or 3). If we choose the coordinates of o to be [1, 0, 0], then the equation of X can be given in the form 2 t0ag(t1, t2) + 2t0ag+1(t1, t2) + ag+2(t1, t2) = 0. (4.3) The transformation is given by the formula 0 x0 = −x0ag+1(x1, x2) − ag+2(x1, x2), x1 = x1(ag(x1, x2)x0 + ag+1(x1, x2)),

x2 = x2(ag(x1, x2)x0 + ag+1(x1, x2).

In afﬁne coordinates z1 = x2/x1, z2 = x0/x1 it is given by the formula

−ag+1(1, z1)z2 − ag+2(1, z1) (z1, z2) 7→ z1, . ag(1, z1)z2 + ag+1(1, z1) The base locus consists of 2g + 2 Weierstrass points on X and the singular point p of X taken with multiplicity g + 1. We check 2 2 d2 = (g + 2) − (2g + 2) − (g + 1) = 1. The P -locus consists of 2g + 2 lines joining o with one of the Weierstrass points and the curve po(X) of degree g + 1 that passes through the Weierstrass points and intersects X at o with multiplicity g(g + 1). 4.1. DE JONQUIERES` INVOLUTIONS 77

Note that the De Jonquieres` involution associated to a submonoidal hypersurface X has X in the closure of its fixed points. As we discussed in the beginning of this section, there are two involution on each line `x. One fixes the intersection point of `x with X and another interchanges them. This allows us to define another Cremona involution that commutes with the De Jonquieres` involution. It is not defined uniquely. We assume that X is given by equation 2 t0am−2(t1, . . . , tn) + 2t0am−1(t1, . . . , tn) + am(t1, . . . , tn) = 0. (4.4) and look for the hypersurface Y in the form 2 t0bm0−2(t1, . . . , tn) + 2t0bm0−1(t1, . . . , tn) + bm0 (t1, . . . , tn) = 0. (4.5) The singular point p = [1, 0,..., 0]. A line up + vx = 0 passing through a point x 6= p intersects X (resp. Y ) at two residual points satisfying the quadratic equation 2 u am−2(x1, . . . , xn) + 2uvam−1(x, . . . , xn) + am(x1, . . . , xn) = 0 (resp. 2 u bm0−2(x1, . . . , xn) + 2uvbm0−1(x, . . . , xn) + bm0 (x1, . . . , xn) = 0) The condition that the two pairs are harmonically conjugate is 0 am−2bm0 − 2am−1bm0−1 + ambm0−2 = 0. (4.6) If we choose Y satisfying this condition and define the corresponding De Jon- quieres` involution T 0, then this involution will restrict to an involution on X. To find the coefficients bm−2, bm−1, bm satisfying (4.6), we have to solve a system of linear equations. Obviously, a solution is an element of the kernel of a linear map

C[t1, . . . , tn]m0 ⊕ C[t1, . . . , tn]m0−1 ⊕ C[t1, . . . , tn]m0−2 → C[t1, . . . , tn]m+m0−2. We expect to find a non-trivial solution when m0+n−1 m0+n−2 m0+n−3 m+m0+n−3 n−1 + n−1 + n−1 > n−1 . For example, when n = 2, the inequality is 2m0 > m. Note that the two commuting Cremona involutions T and T 0 defined by two submonoidal hypersurfraces whose equations are related by (4.6) define the third involution T 00 = T ◦ T 0. One can show that it is defined by the third submonoidal hypersurface of degree m + m0 with equation   am−2 am−1 am det bm0−2 bm0−1 bm0  = 0. 2 1 −t0 t0 It describes the locus of common harmonically conjugate pairs of points. 78 LECTURE 4. INVOLUTIONS

4.2 Planar Cremona involutions

n n Recall that a Cremona transformation T : P 99K P is called an involution if T = T −1. All involutions in the planar Cremona group have been classiﬁed (up to conjugacy). We have already see one example of an involution, the standard Cremona transformation

Tst :[x0, x1, x2] 7→ [x1x2, x0, x1, x0x1].

It is generalized to all dimensions. In affine coordinates, this transformation is −1 −1 −1 1 given by (z1, z2) 7→ (z1 , z2 ). The Moebius transformation z 7→ z of P z−1 is conjugate to the transformation z 7→ −z (by means of the map z 7→ z+1 ). This shows that the standard Cremona transformation Tst is conjugate in Cr(2) to a transformation (z1, z2) 7→ (−z1, −z2). In projective coordinates, it is given by (x0, x1, x2) 7→ (x0, −x1, −x2). Another example of a planar involution is a De Jonquieres` involution associated to a plane hyperelliptic curve of degree d. Its set of fixed points in the domain of the definition is an open subset of the hyperelliptic curve. It is easy to see that its birational type is an invariant of conjugacy class of the transformation. So, non birationally isomorphic hyperelliptic curves define non-conjugate involutions. There are two more examples of involutions: a Geiser involution and a Bertini involution which we now define. The classical definition of a Geiser involution is as follows. Fix seven points 2 p1, . . . , p7 in P (maybe infinitely near) in general position (in the sense that no four points are collinear, not all points lie on a conic). The linear system H = |3H − p1 − ... − p7| of cubic curves through the seven points is two-dimensional. Take a general point p and consider the pencil of curves from H passing through p. Since a general pencil of cubic curves has 9 base points, we can define γ(p) as the ninth base point of the pencil. Another way to see a Geiser involution is as follows. The linear system H defines a rational map of degree 2

2 ∗ ∼ 2 f : P 99K |H| = P . The points p and γ(p) lie in the same fibre. Thus γ is a birational deck transformation of this cover. Blowing up the seven points, we obtain a Del Pezzo surface S 2 of degree 2, and a regular map of degree 2 from S to P . The Geiser involution γ becomes an automorphism of the surface S. It is easy to see that the fixed points of a Geiser involution lie on the ramification curve of f. When the points p1, . . . , p7 are in general position, this curve is a curve of geometric genus 3, of degree 6 with double points at the points p1, . . . , p7. It is 4.2. PLANAR CREMONA INVOLUTIONS 79 birationally isomorphic to a canonical curve of genus 3. In this way, one can see that a Geiser involution is not conjugate to any De Jonquieres` involution. The degree of a Geiser involution is easy to compute. A general line is mapped by the linear system |3H − p1 − ... − p7| to a cubic curve. The pre-image of this cubic curve consists of the line and a curve of degree 8, the image of the line under the Geiser involution. Thus the degree of the Geiser involution is equal to 8. One can show that the homaloidal linear system consists of curves of degree 8 with triple points at p1, . . . , p7.

To define a Bertini involution one considers the linear system of plane sextics with 8 double points p1, . . . , p8. We assume that the linear system of such sextics has no fixed components. Then one can show that it defines a degree 2 map of the 3 blow-up surface Blp1,...,p+8 onto a quadric cone in P . The branch divisor is cut out by a cubic surface. If the points are in general position, then the curve is a smooth curve of genus 4 with a vanishing theta characteristic. The deck transformation of this cover is a Bertini involution. The compute the degree of a Bertini involution, we restrict the double cover to a general line. One can show that its image is a singular curve of degree 6 cut out by a cubic. Its pre-image is equal to the union of the line and a curve of degree 17. This shows that the degree of the Bertini involution is equal to 17. One can show that the homaloidal linear system consists of curves of degree 17 passing through the points p1, . . . , p8 with multiplicities 6. One can also describe ta Bertini involution in the following way. Consider the pencil of cubic curves through the eight points p1, . . . , p8. It has the ninth base point p9. For any general point p there will be a unique cubic curve C(p) from the pencil which passes through p. Take p9 for the zero in the group law of the cubic C(p) and define β(p) as the negative −p with respect to the group law. This defines 2 a birational involution on P , a Bertini involution. One can show that the fixed points of a Bertini involution lie on a canonical curve of genus 4 with vanishing theta characteristic (isomorphic to a nonsingular intersection of a cubic surface 3 and a quadric cone in P ). So, a Bertini involution is not conjugate to a Geiser involution or a De Jonquieres` involution. It can be realized as an automorphism of the blowup of the eight points (a Del Pezzo surface of degree 1), and the quotient 3 by this involution is isomorphic to a quadratic cone in P . The following theorem due to E. Bertini shows that there is nothing else.

Theorem 4.2.1. Any element of order 2 in Cr(2) is conjugate to either a projective involution, or a De Jonquieres` involution, or a Geiser involution, or a Bertini involution. 80 LECTURE 4. INVOLUTIONS

4.3 De Jonquieres` subgroups

Let K be a subﬁeld of C(z) = C(z1, . . . , zn) such that C(z) is a pure transcendental extension of K. Consider the subgroup AutC(C(z),K) of CrC(n) which leave the subﬁeld K invariant. Then we have an exact sequence of groups

1 → AutK (C(z)) → AutC(C(z),K) → AutC(K) → 1. Since any automorphism of K can be uniquely extended to an automorphism of C(z). This sequence splits and deﬁnes an isomorphism ∼ AutC(C(z),K) = AutK (C(z)) o AutC(K).

In the case when K is pure transcendental extension of C of algebraic dimension k, the subgroup AutC(C(z),K) of AutC(C(z)) is called a De Jonquieres` subgroup of level k. Geometrically, the inclusion K,→ C(z) defines a rational dominant map n P 99K X, where K is the field of rational functions of X. Its general fibre is a rational variety over K. The birational automorphisms from AutC(C(z),K) preserves this rational fibration. n n A Cremona transformation T : P 99K P is a called a De Jonquieres` trans- n−k formation if there exists a rational fibration with generic fibre isomorphic to P that is left invariant by T . We have already seen examples of De Jonquieres` transformations of order 2. It follows from the proof of Proposition 2.8.1 that a dilated transformation is a De Jonquieres` transformation of level n − 1. n−1 n Let us see when an involution in P can be dilated to an involution in P . n−1 n−1 Suppose T : P 99K P be an involution given by homogeneous polynomials G = (G1,...,Gn) of degree d in variables t1, . . . , tn. We are looking for a transformation T˜ given by (F0, QG1, . . . , QGn), where F0(t0, . . . , tn) is a monoidal polynomial of degree d with multiplicity ≥ d−1 at the point [1, 0,..., 0] and Q is a monoidal polynomial of degree r with singular point of multiplicity 2 ≥ r − 1. Write F0 = t0A1 + A2,Q = t0B1 + B2. The transformation T˜ is given by

(F0(F0, QG1, . . . , QGn), (QG1)(F0, QG1, . . . , QGn),..., (QGn)(F0, QG1, . . . , QGn))

= (F0A1(G) + QA2(G), QG˜ 1(G),..., QG˜ n(G)), 2 where Q˜ = B1(G)F0 + B2(G)Q. Since T is the identity, we obtain

Gi(G1,...,Gn) = tiP, i = 1, . . . , n, 4.4. LINEAR SYSTEMS OF ISOLOGUES 81

2 where P (t1, . . . , tn) is a homogeneous polynomial of degree d −1. Now, we must have F0A1(G) + QA2(G) = t0(B1(G)F0 + QB2(G))P.

Comparing the coefﬁcients at t0, we ﬁnd

A1(G)A1 + A2(G)B1 = B1(G)A2P + B2(G)B2P,

B1(G)A1 + B2(G)B1 = 0. For example, if Q = 1, we must have

A1(G1,...,Gn)A1 = P. (4.7)

Example 4.3.1. Suppose T = Tst is the standard Cremona transformation of degree n−1 d = n − 1 in P . Then Gi = t1 ··· tn/ti, i = 1, . . . , n. It is easy to see that n−2 n−2 P = (t1 ··· tn) . Thus, if we take A1 = t1 , we obtain (4.7). Therefore, n−2 n taking F0 = t0t1 , we obtain a Cremona involution of degree n − 1 in P . It is given by the polynomials

n−2 (t0t1 , t2 ··· tn, . . . , t1 ··· tn−1).

4.4 Linear systems of isologues

n n Let f : P 99K P be a Cremona transformation of algebraic degree d and Γf ⊂ n n n P × P be its graph. Consider the natural rational map ι :Γf 99K G1(P ) to the n Grassmann variety of lines in P which assigns to a point (x, f(x)) ∈ Γf the line hx, f(x)i spanned by the points x and f(x). It is not deﬁned on the intersection of Γf with the diagonal ∆. Let C(f) be the closure of f(Γf \ ∆). We call it the complex of lines associated to f. Note that

C(f) = C(f −1).

We say that f is non-degenerate if C(f) is not contained in a proper subspace n 2 n+1 of P . In terms of Plucker¨ coordinates pij in Λ (C ), the complex Kf has a n rational parameterization φ : P 99K C(f), given by

pij = tiFj(t0, . . . , tn), 1 ≤ i < j ≤ n.

n The map φ is equal to the composition of the inverse of the projection Γf → P n and the map ι. If the closure of fixed points in dom(f) is a hypersurface in P of degree k, then this hypersurface is a fixed component of each isologue. After we subtract this hypersurface from the linear system, we obtain a linear system of 82 LECTURE 4. INVOLUTIONS hypersurfaces of degree d + 1 − k without fixed components. It is called the linear system of isologues. Its members are isologues. If f is non-degenerate, then the linear system of isologues is parametrized by n the dual of the Plucker¨ space. If P = |E| for some vector space E of dimension n + 1, then isologues are elements of |Λ2E∨|. An isologue is called special if it ∨ ∨ belongs to the Grassmannian G1(|E |) of lines in |E |. Identifying a line in the n n dual P with a codimension 2 subspace Π in P , we see that the corresponding isologue hypersurface VΠ is equal to the closure of the set

n {x ∈ P : x 6= f(x), hx, f(x)i ∩ Π 6= ∅}.

In other words, a special isologue is the pre-image of a hyperplane in |Λ2E| which cuts out the Grassmannian along the special Schubert variety hΠi of lines inter- ∨ secting Π. For example, when n = 2, we can identify G1(|E |) with |E|, hence all isologues are special. They are parameterized by points p ∈ |E| and equal to the closures of the sets

2 {x ∈ P : f(x) 6= x, p ∈ hx, f(x)i}.

If f is degenerate, then some of these loci may coincide with the whole plane. A Cremona transformation f is called Arguesian if dim C(f) < n. This n 2 implies that the rational map φ : P 99K G1(P ) has fibres of positive dimension. Generically, the fibres are lines hx, f(x)i, and the transformation leaves these lines invariant. In the case n = 2, this is equivalent to that f is degenerate. It is clear that any Arguesian transformation is of De Jonquieres` type, i.e. belongs to some De Jonquieres` group. The De Jonquieres` involutions defined by a submonoidal hypersurface are Arguesian. If n = 3, then f is Arguesian if C(f) is an irreducible congruence1 of lines in 3 3 G1(P ). Recall that G1(P ) is isomorphic to a nonsingular quadric in the Plucker¨ 5 space P . It contains two families of planes:

3 P (x) = {lines containing a point x ∈ P }, 3 P (π) = {lines contained in a plane in P }.

The cohomology classes of these planes freely generate

4 3 ∼ 3 ∼ 2 H (G1(P ), Z) = A2(G1(P )) = Z .

3 The class of any congruence S in G1(P ) is determined by two numbers (m, n), called the order and class. The order is equal to the number of lines in S passing

1codimension 2 subvariety in the Grassmannian 4.5. ARGUESIAN INVOLUTIONS 83

3 through a general point in P . The class is equal to the number of lines in S lying 3 3 in a general plane in P . Since through a general point in P passes only one fibre of the map φ, we see that the image of φ is a congruence of order 1. n Example 4.4.1. Consider the standard Cremona transformation Tst in P . Its graph n is the toric variety X(An). The transformation has 2 fixed points with affine coordinates zi = ±1. The linear system of isologues is a linear system of hypersurfaces of degree n+1 passing through the fixed points and the base locus of Tst. If n = 2, then X(A2) blown up at 4 points is a weak Del Pezzo surface of degree 7 (weak means that the anticanonical class is nef and big but not ample). The map ι is a rational map of degree 2 onto the plane. The branch divisor is the union of 4 lines in a general position. If n = 3, then isologues are quartic surfaces which contain the edges of the coordinate tetrahedron and pass through 8 fixed points of Tst. The linear system of 3 isologues is 4-dimensional. It maps X(A3) onto a hyperplane section of G1(P ), a linear complex of lines. It is isomorphic to a 3-dimensional quadric. This shows that f is degenerate but not Arguesian . The degree of the map is equal to 6. Special isologues are quartic which are singular outside the vertices of the tetrahedron. Remark 4.4.1. The linear system of isologues is not an invariant of the homaloidal linear system. It depends on the map itself, i.e. a choice of a basis in the homaloidal linear system. In the previous example, say n = 3, if switch the first two coordinates, we find that the fixed locus consists of 4 lines: V (t0, t2 ± t3) and V (t1, t2 ± t3).

4.5 Arguesian involutions

n We know that there exist Arguesian involutions in P of any degree d. For example, De Jonquieres` involutions occur in every degree. They leave invariant a (n − 1)-dimensional variety of lines passing through a ﬁxed point. Here we will 3 construct Arguesian involutions of P of degree d = m + 5. which leaves invariant a congruence of lines of bidegree (1, m + 1). The following proposition is a result of E. Kummer (see a modern proof in [Z.Ran, Crelle Journa, 1984]).

Proposition 4.5.1. Let γ1 be a line and γ2 be a smooth rational curve of degree m + 1 intersecting γ1 transversally at m points. The closure of the set of lines intersecting γ1 and γ2 at one point consists of the union of m congruences of bidegree (1, 0) formed by lines passing through one of the intersection points and an irreducible congruence of bidegree (1, m + 1). Any irreducible congruence of lines of bidegree (1, m) is either obtained in this way, or is equal to the congruence of secant lines of a skew cubic (in this case m = 3). 84 LECTURE 4. INVOLUTIONS

3 Let S ⊂ G = G1(P ) be an irreducible congruence of lines obtained from a reducible curve γ1 ∪ γ2 as above. Note that all lines in this congruence intersect γ1, so the congruence is contained in the tangent hyperplane of G at the point corresponding to γ1. In the case m = 0, it is a 2-dimensional quadric contained in another hyperplane. Take any point x on the line γ1. The set of lines from S which pass through x is a ruling of a cone of degree m + 1 with the vertex at x. In S, this defines a linear pencil C(x), x ∈ `1, of curves of degree m + 1 parameterized by `1. If m > 0, the line γ1 itself belongs to all such cones, it is a base point s0 of this pencil. Now, we take any point on γ2 and do the same. This time we obtain a pencil of lines L(x), x ∈ γ2, in G, parameterized by γ2. A curve C(x), x ∈ γ1, intersects a line L(y), y ∈ γ2, at one point represented by the line hx, yi. Consider a rational map ψ : γ1 × γ2 99K S that sends a point (x, y) to the intersection point C(x) ∩ L(y). It is easy to see that the map is defined everywhere except at the intersection points (xi, yi), xi = yi. Thus it extends to a regular map from γ1 × γ2 to S. The intersection points go to the same point s0. This shows that −1 ψ is a normalization map, and #f (s0) = m. When m > 0, the surface S is a non-normal scroll with an isolated singular point. The map f is given by a linear system of dimension 4 contained in the linear system |l1 + (m + 1)l2|, where |l1| and |l2| are the rulings. So, S is a projection of F0, embedded as a scroll Σ0,2m+2 2m+3 4 of degree 2m + 2 in P , to P . Suppose Γ is contained in a smooth quartic surface X. It follows from the theory of periods of K3 surfaces that quartic surfaces containing some Γ as above form a subvariety of codimension 2 in the projective space of quartics. Consider 3 the following Cremona involution T in P . Take a general point x, pass a secant `x of Γ through x which intersects Γ at two points a and b. Define the involution on `x by designating a and b to be its fixed points. Then T (x) is defined to be conjugate to x under this involution. Let us locate the base points of T . Obviously, the map is not defined at the curve γ1 ∪ γ2. It is also not defined at the points where the secant line ` is tangent to X. Everywhere else the map is well defined. Let us determine the locus R of the tangency points. Consider the birational involution τ of X that assigns to a point x ∈ X the intersection point of the secant `x of Γ which is not contained in Γ. A birational involution of a K3 surface extends to a biregular involution. The conjugate points are on a line from the congruence S, so the fixed locus of the involution is the curve R we are looking for. The linear system of planes through the line γ1 cuts out on X a pencil of elliptic cubic curves. It is easy to see that τ induces th involution on each cubic curve contained in a plane Π obtained by projection from the unique point in Π which lies on γ2 but does not lie on γ1. In particular, the divisor class 4.5. ARGUESIAN INVOLUTIONS 85

F = H − γ1 is τ-invariant. 3 Consider the linear system in P

H = |(m + 2)H − (m + 1)γ1 − γ2|.

It is clear that any line from the congruence S is blown down to a point. The dimension of H is easy to compute. A homogeneous form of degree m + 2 vanishing on a line V (t0, t1) with multiplicity ≥ m + 1 is equal to a linear com- i m+2−i i m+1−i i m+1−i bination of monomials t0t1 and monomials t0t1 t2, t0t1 t3. Thus the dimension of |(m + 2)H − (m + 1)γ1| = 3m + 6. To contain γ2 requires (m + 1)(m + 2) + 1 − m(m + 1) = 2m + 3 additional conditions. This gives dim H = m + 3. Let |D| = |(m + 2)h − (m + 1)γ1 − γ2| be the restriction of H to X. It is a complete linear system and an easy computation gives D2 = 2m + 4. It is well- known that a complete linear system on a K3 surface without fixed components has m+3 no base points. Thus the linear system defines a regular map f : X → P onto a surface of degree 2(m + 4)/ deg f. The linear pencil |E| = |h − γ1| of cubic curves defines an elliptic fibration on X with E · D = 2. This shows that the linear system |D| is a hyperelliptic linear system on X and the map f is of degree 2 on a m+3 scroll of degree m + 2 in P . Our curve R is the ramification curve of the map f. Assume that X is general in the sense of periods of K3 surfaces. This means that the divisor classes h, γ1 and γ2 generate Pic(X). The involution τ defined ⊥ 2 by the double cover f acts on the transendental lattice Pic(X) ⊂ H (X, Z) as the minus identity. It has two invariant divisor classes D and H − E. Since D = (m + 1)(H − γ1) + (H − γ2) we may assume that the invariant part of Pic(X) is generated by H − γ1 and H − γ2. We have

2 2 (H − γ1) = 0, (H − γ2) = −2m, (H − γ1) · (H − γ2) = 2.

Since R is invariant with respect to τ, we can write

R ∼ a(H − γ1) + b(H − γ2). (4.8)

Since any nonsingular fibre of the elliptic fibration |H − γ1| has 4 fixed points, we get R · (H − γ1) = 4. Intersecting both sides of the previous equality with H − γ1 we obtain b = 2. Since R2 = 16, we obtain 16 = 4ab − 2mb2 = 8a − 8m. This gives a = m + 2. Now

deg R = R · H = 3(m + 2) + 2(4 − m − 1) = m + 12. 86 LECTURE 4. INVOLUTIONS

To summarize, we obtained that the base locus of T consists of a γ1, γ2 and a smooth curve R of degree m + 12 and genus 9. Also observe that, using (4.8) with a = m + 2, b = 2, we obtain that

R · γ1 = m + 8,R · γ2 = 3m + 8.

Let us determine the multiplicities of the base curves. Take a general point x ∈ γ2 and consider the plane Π = hx, γ1i. The transformation T leaves this plane invariant and fixes pointwisely the cubic curve E = Π∩X −γ1. We know also that the involution τ of X induces on E the involution defined by the projection from x. We immediately recognize that T |Π is a De Jonquieres` involution of degree 3 whose homaloidal linear system consists of cubic curves singular at x and having simple base points at the 4 = (m + 12) − (m + 8) intersection points of R with E outside of γ1. This implies that the curve γ1 is a curve of multiplicity d − 3 for a general divisor from the homaloidal linear system defining T and the curve γ2 is a double curve. The curve R enters with multiplicity 1. Let Q be the image of a general plane H under T . It is a rational surface of degree d equal to the algebraic degree of T . The homaloidal linear system defining T restricts to H to define a linear system of curves of degree d passing through the set Σ = (R ∪ γ1 ∪ γ2) ∩ H. From above we know that any intersection point with γ1 is a point of multiplicity d − 3 and the intersection point with γ1 is a point of multiplicity 2. It has also m + 1 simple points. Since it maps H to a surface of degree d we must have

2 X 2 2 d − d = mi = (d − 3) + 4(m + 1) + 12 + m. pi∈Σ

This gives d = m + 5. One can also describe the P -locus of the involution T . It consists of the ruled surface of degree 8 + 2m = R ·H whose rulings are tangents to R, a quartic surface which is blown down to γ2, and a surface of degree 2m + 4 which is blown down to γ1. Let us consider the case of the congruence S of bidegree (1, 3) of secants of a 5 twisted cubic γ. The surface S, embedded in the Plucker¨ space P is a Veronese surface. Again we choose a general quartic surface X containing γ. The map f : X → S is a double cover branched along a smooth plane sextic curve. It is defined by the linear system |2h − γ|. The ramification curve R belongs to the linear system |6h − 3γ|. Its genus is 10 and its degree is 15. The base curve is the union of the curves γ and R intersecting at 24 points. We skip the details and 4.5. ARGUESIAN INVOLUTIONS 87 leave to the reader to check that the homaloidal linear system defining T consists of surfaces of degree 7 vanishing along γ with multiplicity 3. The P -locus consists of the tangential ruled surface of γ of degree 4 and a ruled surface of degree 12 whose rulings are tangent to R. Example 4.5.1. We will discuss a special classical example of an Arguesian involutions of degree 5. It is obtained from the previous construction when the quartic surface is special and the curve R becomes equal to the union of 8 skew lines and a quartic elliptic curve intersecting each line at two points. 3 Fix a general pencil Q of quadrics in P . The restriction of Q to a general ` ∈ S defines a pencil of degree 2, hence an involution on `. This defines an Cremona 3 3 involution on P . Take a general point x = [v] in P , pass a unique line through x which intersects `1 and `2, and send x to T (x) such that the pair (x, T (x)) is in the involution defined by Q. This is an example of a De Jonquieres` type involution of 1 level 1. Note that, for any filed K, an automorphism of order 2 of PK is conjugate to an automorphism given by (t0, t1) → (t1, at0 + t1) for some a ∈ K. In formulas, let `1 = V (t0, t1), `2 = V (t2, t3). For any x = [a0, a1, a2, a3], the plane hx, `1i is equal to V (−a1t0 +a0t1). The plane hx, `2i is equal to V (−a3t2 + a2t3). The planes intersect along the line ` given in parametric form by [sv1 +tv2], where v1 = (a0, a1, 0, 0), v2 = (0, 0, a2, a3). The pencil of quadrics V (u0q1 + u1q2) restricted to the line ` is equal to the pencil

A(u, a)s2 + 2B(u, a)st + C(u, a)t2 = 0, (4.9) where

A = u0q1(v1) + u1q2(v1),

B = u0b1(v1, v2) + u1b2(v1, v2),

C = u0q1(v2) + u1q2(v2).

Here bi are the symmetric bilinear forms associated to the quadratic forms qi. The coefﬁcients A, B, C are bi-homogeneous functions in the coordinates u = (u0, u1) and a = (a0, a1, a2, a3) of degree 1 in u and degree 2 in a. The point x corresponds to the parameters [s, t] = [1, 1]. The quadratic form has 0 at [1, 1]. This allows one to express the coefﬁcients of equation (4.9) as polynomials of degree 4 in the coordinates ai:

[u0 : u1] = [q2(v1 + v2), −q1(v1 + v2)] = [q2(v), −q1(v)]

This gives the second solution [s, t] of (4.9) as polynomials of degree 4

[s, t] = [C,A] = [q2(v)q1(v2) − q1(v)q2(v2), q2(v)q1(v1) − q1(v)q2(v1)] 88 LECTURE 4. INVOLUTIONS

Finally, the coordinates of the transform T (x) of the point x become expressed as polynomials of degree 5 in the ai’s

T (x) = [a0s, a1s, a2t, a3t].

This shows that the involution T is of algebraic degree 5. Let us see where our transformation is not defined. Obviously, it is not defined on the lines `1 and `2, as well on the base locus E of the pencil of quadrics. Also, it is not defined on any secant of C which intersects `1 and `2. In fact, the restriction of the pencil to this secant consists of two points which do not move. The set of 3 secants of E is a congruence S1 of lines in the Grassmannian G1(P ) of bi-degree (2, 6). The set of lines intersecting `1 and `2 is a congruence S2 of bi-degree (1, 1). They intersect at 8 points. Thus the base locus contains 10 lines and a quartic curve. This can be confirmed by the explicit formulas for the transformation. They also show that the lines `1 and `2 enter with multiplicity 2. Let us find the locus of fixed points of T . A general point x satisfies T (x) = x if and only if [s, t] = [1, 1] is the unique solution of the quadratic equation (4.9). So, we get the equation

A − C = q2(v)(q1(v2) − q1(v1)) + q1(v)(q2(v1) − q2(v2)) = 0.

It is a quartic surface F with equation

q2(t0, t1, t2, t3)(q1(0, 0, t2, t3) − q1(t0, t1, 0, 0))

+q1(t0, t1, t2, t3)(q2(t0, t1, 0, 0) − q2(0, 0, t2, t3)) = 0.

It follows from the equations that the eight skew lines from the congruence S2 lie on the quartic surface. The curve E itself also lies on the surface. Also, if we plug in v1 = 0 (resp. v2 = 0), we get zero. So, the quartic S also contains the lines `1 and `2. Consider the elliptic pencils |H − `1| and |H − `2| cut out by planes through the lines `1 and `2. The linear system |2H − `1 − `2| maps F two-to-one onto a ∼ 1 1 3 nonsingular quadric Q = P × P in P . Of course, this extends the rational map which assigns to x ∈ F the points (`x ∩ `1, `x ∩ `2), where `x is the line in the congruence S1 passing though x. Each pencil is the pre-image of a ruling. Since ì ·(H −`1) = ì ·(H −`2) = 0 for i = 3,..., 10, we see that the eight secants `3, . . . , `10 are blown down to singular points q1, . . . , q8 of the branch curve W . The ramification curve of the double cover is our quartic curve E. It belongs to the divisor class 4H − 2`1 − 2`2 − `1 − ... − `10, where H is a plane section of X. 4.5. ARGUESIAN INVOLUTIONS 89

We have (H−`1)·`1 = 3, (H−`1)·`2 = 1 and (H−`2)·`1 = 1, (H−`2)·`1 = 1. This implies that `1 (resp. `2) is mapped isomorphically to a curve R1 (resp. R2) of bi-degree (3, 1) (resp. (1, 3)). They intersect at q1, . . . , q8 and also at two more points a, b. The pre-image of Ri under the cover S → Q splits into the sum 0 0 of two lines ì +ì with ì ·`j = 1, i 6= j. The images of the two intersection points 0 0 `1 ∩ `2 and `1 ∩ `2 are the intersection points a, b of R1 and R2. 1 1 Conversely, start with a set P = {q1, . . . , q8} of 8 points on Q = P × P such that the dimension of the linear system of curves of bidegree (4, 4) with double points at each qi ∈ P is a pencil containing a reducible curve R1 + R2, where R1,R2 are smooth rational curves (necessary of bidegrees (3, 1) and (1, 3)). One can show that the sets of such points depends on 14 parameters (by projecting from q8, they correspond to base points of an Halphen pencils of elliptic curves of degree 6 with one reducible member equal to the union of two rational cubics). Choose a smooth member W of the pencil and consider the double cover of Q branched over W . Let Q0 be the blow-up of the set P, and π : X → Q0 be the double cover of Q0 ramified along the proper transform W 0 of W . 0 The proper transform Ri of the curves Ri splits into the disjoint union of two 0 0 0 0 (−2)-curves ì + ì such that `1 + `2 ∼ `2 + `1 and ì · `j = 2, i 6= j. Let |fi| the 1 pencils on Q’ defined by the two rulings Q → P . We have ∗ 0 0 π (R1 − R2) = `1 + `1 − `2 − `2 ∼ 2(`1 − `2) ∗ ∗ ∗ ∗ ∼ π (3f1 + f2) − π (f1 + 3f2) = 2π (f1) − 2π (f2). This gives ∗ ∗ π (f1) + `2 ∼ π (f2) + `1. Set ∗ H = π (f2) + `1. 2 2 2 We have H = 2(f2 + R1) + R1 = 6 − 2 = 4. Also H · ì = 1. The linear system 3 |H| maps X onto a quartic surface in P and the images of `1 and `2 are lines. 0 Let `1, . . . , `8 be the pre-images of the exceptional curves Li of Q → Q under π. These are (−2)-curves on X. We have

∗ H · ì = (π (f2) + `1) · ì = `1 · ì = R1 · Li = 1, i = 1,..., 8.

So, the curves `i, i = 3,..., 10, are mapped to lines on the quartic model of X. Finally, ∗ ∗ E ∼ 2(π (f1) + π (f2)) − `3 − ... − `10, and we easily check that H · E = 4, so that E is mapped to an elliptic quartic curve. The lines `3, . . . , `10 are common secants of E and `1 + `2. So, we can 3 reconstruct an involution T of P with ﬁxed locus isomorphic to X. 90 LECTURE 4. INVOLUTIONS

Finally, observe that the set of involutions T depend on 24 parameters, 8 for the pairs of skew lines `1 and `2, and 16 for pencils of quadrics. Modulo projective 3 transformations of P , we have 9 parameters. The isomorphism classes of our quartic surfaces depend on the same number of parameters, 8 for sets of points P and one for a choice of its member. Another class of Arguesian involutions which exist in any projective space of odd dimension are deﬁned by varieties with an apparent double point. First let us recall the following deﬁnition.

n Definition 4.5.1. A closed subvariety X of P is called a subvariety with one n apparent double point if a general point in P lies on a unique secant of X. n A subvariety X of P with an apparent double point (an OADP- variety) defines n a Cremona involution of P in a way similar to the definition of a De Jonquieres` n involution. For a general point x ∈ P we find a unique secant of X intersecting X at two points (a, b), and then define the unique T (x) such that the pair {x, T (x)} is harmonically conjugate to {a, b}. A most general example of a variety with one apparent double points is an Edge variety. The Edge varieties are of two kinds. The first kind is a general 1 n 2n+1 divisor En,2n+1 of bidegree (1, 2) in P × P embedded by Segre in P . Its degree is equal to 2n + 1. For example, when n = 1, we obtain a twisted cubic in 3 5 P . If n = 2, we obtain a Del Pezzo surface in P . The second type is a general 1 n divisor of bidegree (0, 2) in P × P . For example, when n = 1, we get the union 5 of two skew lines. When n = 2, we get a quartic ruled surface in P isomorphic to 1 1 P × P embedded by the linear system of divisors of bidegree (1, 2). Let us recall what is known about smooth varieties X with one apparent double 2n+1 point. First, it is clear that an n-dimensional X spans P .

Theorem 4.5.2. Suppose n = 2, then X is an Edge variety E2,4 or E2,5. If n = 3, 7 then X is either an Edge variety E3,6 or E3,7, or a scroll of degree 8 in P equal to the intersection of a quadric hypersurface with the cone over the Segre variety 1 2 5 s(P × P ) ⊂ P with a line as a vertex. The case n = 2 is classical and goes back to F. Severi, and the case n = 3 was treated in a paper by Ciliberto, Mella, Russso in JAG. The Cremona involution defined by E1,2 (a pair of skew lines) is a projective transformation given (in notation of Example 4.5.1) by [a0, a1, a2, a3] 7→ [a0, a1, −a2, −a3]. The Cremona involution defined by a skew cubic E(1, 3) is the polarity with respect to the net of quadrics containing the cubic. It is a bilinear transformation which we considered in Example 3.4.4. The Cremona involution defined by a Del Pezzo surface X of degree 5 is also the polarity involution with 3 4.6. GEISER AND BERTINI INVOLUTIONS IN P 91

5 respect to the 4-dimensional linear system of quadrics in P . It is a bilinear transformation of degree 5. Its exceptional divisor is the tangential variety of X of degree 24. Its base locus is a double structure on X.

4.6 Geiser and Bertini involutions in P3 The following is a classical generalization of the Geiser involution in the plane. 3 3 Assume n = 3, and consider a rational map φ : P 99K P defined by the linear system H = |2H − p1 − ... − p6|, where p1, . . . , p6 are points in general linear 3 position. Take a general point x ∈ P , then the quadrics from H passing through x form a net. The base locus of the net consists of 8 points, we have already seven base points p1, . . . , p6, x. By definition, T (x) is the eighth base point. A general plane is mapped to a quartic surface, a projection of a Veronese surface. The pre-image of this quartic surface consists of the plane and a surface of degree 7. This shows that the degree of the involution is equal to 7. Let ìj = hpi, pji and R be a a twisted cubic through the six points (it is uniquely defined). The base locus of the involution consists of the union of 15 simple lines ìj and the curve R taken with multiplicity 3. The P-locus consists of the six quadrics from the linear system which have a singular point at one of the points pj. The set of fixed points of the involution is a famous Weddle quartic surface with nodes at p1, . . . , p6. It is the locus of singular points of singular quadrics from the linear system |2H − p1 − ... − p6|. The rational map φ maps this surface birationally onto a Kummer quartic surface K. The 15 lines and the curve R are blown down to the 16 nodes of the Kummer surface. One can show that the complex C(T ) defined by the linear system of isologues is the quadratic line complex associated to the Kummer surface K (see Griffiths- Harris book, or Topics). Remark 4.6.1. The Geiser involution can be also defined using the geometry of the 4 Segre cubic primal S3, a cubic hypersurface in P with 10 nodes. It is known that 3 the linear system |2H−p1−...−p6| defines birational isomorphism ψ : P 99K S3. It blows down the 10 chords hpi, pji to the singular points of S3. Consider the 3 projection pr : S3 99K P from a nonsingular point on S3. The composition 3 3 3 3 p ◦ ψ : P 99K P is a degree 2 rational map P 99K P . The deck transformation is the Geiser involution. 3 The planar Bertini involution can be generalized to an involution in P as fol- 3 lows. Consider a net of quadrics in P passing through 7 general points p1, . . . , p7. 3 It has additional base point o. Take a general point x ∈ P . The quadrics from the net passing through this point define a pencil P(x). Its base locus is an elliptic curve E(x) of degree 4 with a fixed point o. We use it to define a group law on 92 LECTURE 4. INVOLUTIONS

E(x). We set T (x) to be equal to −x in the group law. This is the definition of a space Kantor involution due to A. Coble. One can rephrase it in modern terms as follows. Let X be the blow-up of the set {p1, . . . , p7, o}. The linear system 2 |2H − p1 − ... − p7 − o| defines an elliptic fibration f : X → P with a dis- −1 tinguished section Eo = σ (o). The open subset of nonsingular points of the 2 fibres is a group scheme over P . The Kantor involution restricted to this open set is the negation morphism x 7→ −x. The fixed locus of the Kantor involution is the union of the section Eo and a surface S of degree 6 with triple points at pi’s. The 2 restriction of f to the proper transform S¯ of S in X is a degree 3 cover of P . Its branch locus is the dual of a plane quartic curve. All of this is well documented in Coble’s book “Algebraic geometry and theta functions” and also in my book with D. Ortland (Asterisque, vol. 168). Consider the linear system H of quartic surfaces with double points p1, . . . , p7. Its dimension is equal to 34 − 7 · 4 = 6. Besides the seven double points, the base locus contains a curve C of degree 6 and genus 3, the locus of singular points of the elliptic fibration f. Let Qi = V (qi), i = 1, 2, 3, be a basis of the net of quadrics through these points. For any quadratic form Q in three variables, the 0 quartics V (Q(q1, q2, q3)) form a 5-dimensional subsystem H of H. The whole linear system is generated by this subsystem and some quartic V (F4). The linear 6 system defines a regular degree 2 map π : Blp1,...,p7 → P . Its image Y is the cone 5 over a Veronese surface in P with the vertex equal to the image of the point o. The branch divisor is cut out by a cubic threefold. This is in a complete analog with 3 the planar Bertini involution. In this way, the quotient of P becomes birationally isomorphic to a rational variety Y . One can compute the degree of the Kantor involution as follows. The restriction of the linear system H to a general plane is a 6-dimensional linear system of quartic 0 5 curves. The linear system H maps it 4 : 1 to a Veronese surface in P . This shows that the projection of the image of the plane from the vertex of the cone is the intersection of the cone with a surface of degree 4. As in the Geiser involution, this implies that the degree of the Kantor involution is equal to 15. The locus of fixed points of a general Kantor involution is surface of degree 6 with triple points at p1, . . . , p7. The base locus consists of the union of 21 chords 0 hpi, p7i and the union of 7 skew cubics through 6 points pis. One can generalize a Geiser involutions as follows. A starting point is an observation that the base ideal JZ of the linear system |3H − p1 − ... − p7| is given by a resolution 3 0 → OP2 (−1) ⊕ OP2 (−2) → OP2 (−3) → JZ → 0. 2 This follows from the fact that the scheme of 7 points is a ACM subscheme of P of codimension 2. Arguing as in the case of biregular transformations, we deduce 3 4.6. GEISER AND BERTINI INVOLUTIONS IN P 93

3 3 from this that the graph of the transformation is a complete intersection in P × P 2 2 of divisors of type (1, 1) and (2, 1). It defines a map of degree 2 from P to P . The Geiser involuitiuon is the deck transformation of this map. We use this observation n n to define a degree 2 map P 99K P whose graph is a complete intersection of n−1 divisors of type (1, 1) and one divisor of type (2, 1). The linear system defining this map has the base ideal given by a resolution

n−1 n+1 0 → OPn (−1) ⊕ OPn (−2) → OPn (−n − 1) → JZ → 0. Example 4.6.1. Take n = 3. Then, computing the Hilbert functions, we find that the base scheme Z is a curve of degree 11 and genus 14. For any smooth quartic S containing Z, a residual curve C in |4h − Z| is a curve of genus 2 and degree 5. 2 The linear system |4h − Z| defines a degree 2 map from S onto P . It is branched along a curve of degree 6. The linear system |2h − C| consists of a skew cubic R. Conversely, a quartic surface containing a skew cubic R defines the linear system |2h − R| of curves of degree 5 and genus 2, and the linear system |2h + R| consists of curves of degree 11 and genus 14. 3 3 Let X = BlZ P → P be the degree 2 regular map defined by the linear system 3 |4H − Z|. The branch divisor is a surface B of degree 6 in P . The ramification divisor R is the proper transform of a surface of degree 12 which passes through Z with multiplicity 3. The exceptional divisor E is a ruled surface isomorphic to ∼ 3 P(OZ ⊕ OZ (4h − C)) = F18. It is mapped by φ onto a ruled surface in P whose rulings are tritangent lines of the sextic surface B. Also, it follows from the Cayley formula for the number of quadrisecant lines of a curve of genus g and degree d

(d − 2)(d − 3)2(d − 4) (d2 − 7d + 13 − g)g t = − , 4 12 2 that Z has 35 quadrisecant lines (see [Grifﬁths-Harris]). They are blown down to ordinary double points of B. 3 Note that the double cover of P branched along a general surface of degree 6 is a non-rational variety. In our case, it is a rational variety. Its intermediate Jacobian variety is isomorphic to the Jacobian variety of the curve Z. Finally, observe that the Geiser involution of this type is not conjugate to the Geiser involution described in above since the surfaces of ﬁxed points are not birationally isomorphic. 94 LECTURE 4. INVOLUTIONS Lecture 5

Factorization Problem

5.1 Elementary links

In this lecture we assume that the reader is somewhat familiar with some basics of the Minimal Model Program. Let M be the full category of the category of projective algebraic varieties whose objects have terminal Q-factorial singularities. 1 We denote by N (X) the vector space Num(X)⊗R, where Num(X) is the Picard group modulo numerical equivalence. For any projective morphism U → V , we denote the relative Mori cone by NE1(U/V )) ⊂ N1(U/V ). It is the real closure in N1(U) = A1(U) ⊗ R of the cone of effective curves contained in ﬁbres. The nef 1 1 cone is the dual cone NE (U/V )) ⊂ N (U/V ) = Pic(U) ⊗ R. The real closure 1 of the cone of effective divisors is denoted by NM (U). Its elements are called quasi-effective divisor classes.

Definition 5.1.1. A Mori fibre space is a projective morphism φ : X → S, where X ∈ M and S is a normal variety with dim S < dim X. Its fibres are connected, −KX is relatively ample (i.e. for any curve C in fibres KX · C < 0) and dim N 1(X) = dim N 1(S) + 1. A morphism of fibre spaces is a diagram

f X / X0

φ φ0 SS0 where f is a birational map. There is no arrow at the bottom, so we keep the vertical arrows to remember that we have structures of Mori spaces on X and X0. A morphism of Mori ﬁbre spaces is called a square (resp. Sarkisov isomor-

95 96 LECTURE 5. FACTORIZATION PROBLEM

0 phism) if, additionally, there is a rational map g : S 99K S such that the diagram

f X / X0

φ φ0 g S / S0 is commutative (resp. it is square and f is an isomorphism)

It is known that a Mori fibre space is obtained by contraction of an extremal ray in NE(X). 2 In dimension 2, a Mori fibre space is either a constant morphism of P , or a projective bundle over a nonsingular curve. In dimension 3, a Mori fibre space is one of the following three kinds:

(i) X → S = point, where X is a Fano variety and dim N 1(X) = 1;

(ii) a conic bundle X → S over a normal surface S with rational singularities, the map contracts an extremal ray represented by an irreducible component of a singular fibre or a fibre if the map is smooth. Each singular fibre is either 1 irreducible, non-reduced, or a bouquet of two P ’s. (iii) a Del Pezzo fibration X → S, where S is a nonsingular curve and a general fibre is a Del Pezzo surface of degree d 6= 8, or a quadric, or a plane. In the second case a singular fibre is a quadric with an isolated singular point. In 2 the last case the fibration is a P -bundle. The map contracts an extremal ray represented by the class of a line on a general fibre.

We will need a log-version of a Mori fibre space. Let (X, ∆) be a log pair that consists of a normal variety X and a Q-divisor on ∆. We assume that KX +∆ is Q- Cartier. For every divisorial valuation v : K → Z of the field of rational functions K on X there exists a birational morphism ν : Z → X of normal varieties and an irreducible divisor E on Z such that the valuation is defined by the discrete valuation of K with the ring of valuation equal to OZ,η, where η is a generic point −1 ∗ of E. The difference KX + ν (∆) − ν (KX + ∆) is a linear combination of exceptional divisors of ν. The coefficient at E is called the discrepancy of ∆ and is denoted by aE(X, ∆). It depends only on the valuation. We say that (X, ∆) has terminal (canonical, log-canonical) singularities if, for all ν : Z → X, the discrepancies aE(X, ∆) are positive (non-negative, ≥ −1). In fact, to check the definitions, it suffices to look at the valuations defined by irreducible exceptional divisors in a log-resolution of the pair (X, ∆). If one takes ∆ = 0, we obtain the definition of terminal, canonical, log-canonical singularities. 5.1. ELEMENTARY LINKS 97

One defines a Mori log fibre space f :(X, ∆X ) → (S, ∆S) by requiring that f(∆X ) = ∆S, the pair (X, ∆) has terminal singularities, X is Q-factorial, ρX/S = 1, and −(KX + ∆X ) is relatively ample. The importance of Mori fibre spaces is explained by the following main result of the Minimal Model program in dimension 3.

Theorem 5.1.1. Let (X, ∆) be a Q-factorial terminal log-pair. Then a sequence 0 of elementary birational transformations X 99K X1 99K ... 99K XN = X , called divisorial contractions and ﬂips, creates a log pair (X0, ∆) with terminal Q-factorial singularities such that one of the the following two alternatives occurs: 0 (i) (X , ∆) is a minimal model, i.e. KX0 + ∆ is nef.

0 0 (ii) (X , ∆) admits a structure of a Mori log space f :(X , ∆X0 ) → (S, ∆S). The goal of Sarkisov’s Program is to factor any birational morphism between two Mori ﬁbre spaces into a composition of elementary links, i.e. morphisms which elementary in the sense we will make precise. Before we describe elementary links let us introduce one more deﬁnition.

0 Definition 5.1.2. A birational map f : X 99K X is called small if there exists an open Zariski subset U whose complement is of codimension ≥ 2 such that the restriction of f to U is an isomorphism. Note that there are no small Cremona transformations of degree > 1 since the P-locus is always a hypersurface. Also any small birational map of normal surfaces must be an isomorphism. However in dimension ≥ 3 small birational maps appear frequently. Definition 5.1.3. A flip (resp. flop) is a diagram of birational maps

f X / X+ ,

φ φ+ } Y where φ and φ+ are birational morphism onto a normal varieties with exceptional loci of codimension ≥ 2. Moreover, KX is relatively φ-ample (resp. numerically + trivial) and −KX+ is relatively φ -ample (resp. numerically trivial). It is clear from this definition that a flip is a small birational map. Following Miles Reid, a birational transformation which is a flip or a flop, or its inverse, will be called a flap. We will be often using the following well-known fact. 98 LECTURE 5. FACTORIZATION PROBLEM

n k Lemma 5.1.2. Let X ⊂ P × P be the graph of the projection map from a linear n subspace L of dimension n − k − 1. Then the first projection map p1 : X → P n k is isomorphic to the blow-up BlLP with exceptional divisor L × P . The second k ∼ projection p2 : X → P is isomorphic to the projective bundle P(E), where E = ⊕n−k O ⊕ O k (1). Pk P Proof. The first assertion follows from the fact that the normal bundle of L is ⊕k isomorphic to OL(1) . The variety X is a complete intersection of k of divisors ∼ ∗ of type (1, 1). Let O(1) = p1OPn (1) be the tautological line bundle corresponding n k k ∼ to the projection pr2 : P ×P → P . Then X = P(E), where E = (p2)∗(OX (1)). Assume first that k = 1. Then we have an exact sequence

0 → OPn×P1 (−1, −1) → OPn×P1 → OX → 0.

∗ Twisting by p1OPn (1) and taking the direct image, we obtain an exact sequence

n+1 0 → O 1 (−1) → O → E → 0. P P1

Then E splits into the direct sum of n line bundles OP1 (ai) with ai ≥ 0. Taking the P ∼ n−1 determinants, we get ai = 1. This gives E = O 1 ⊕ O 1 (1). Assume k > 1. P P It is easy to see that E is a uniform vector bundle, i.e. the isomorphism class of the restriction of E to any line does not depend on the line.1 This implies that E splits into the direct sum of vector bundles OPk (ai). The restriction of the projection n k n 1 pr2 : P 99K P over ` is isomorphic to the projection of P 99K P from the space ∼ n−k+1 hL, `i = P with center at L. Since the restriction of E = ⊕OPk (ai) to ` is ⊕n−k ∼ n−k isomorphic to O ⊕ O`, we obtain that E = O k ⊕ O k (1). ` P P

4 Example 5.1.1. Let Q be a quadric in P with isolated singular point o. We may consider Q as a cone with vertex at o over a nonsingular quadric Q0 in a hyperplane H. The quadric Q has 2 rulings by planes arising from the two rulings P1 and P2 of Q0 by lines. Consider the incidence variety

Xi = {(x, P ) ∈ Q × Pi : x ∈ P }.

Let φi : Xi → Q be the projections to Q. Then φi is an isomorphism over the ∼ 1 complement of the point o, and its ﬁbre Ri over o is equal to Pi = P . We claim −1 that the birational transformation f = φ2 ◦ φ1 : X1 99K X2 is a ﬂop. First, 2 observe that the second projection Xi → Pi is a P -bundle. It corresponds to the 1 composition Xi \ Ri → Q \{0} → Q0 → P , where the last map corresponds to

1[Okonek, Vector bundles on complex projective spaces], Theorem 3.2.3 5.1. ELEMENTARY LINKS 99

4 1 the rulings Pi on Q. The variety Xi embeds in P × P as a complete intersection of two divisors of type (1, 1). We have an exact sequence

0 → OΓ(−1, −1) → OΓ → OXi → 0,

4 1 where Γ is the graph of the projection P 99K P from the point o. Twisting by ∗ 1 p1(OP4 (1) and taking the direct image under p2 :Γ → P , we get, by applying Lemma ??, an exact sequence

⊕3 0 → O 1 (−1) → O ⊕ O 1 (1) → E → 0. P P1 P ∼ ∼ ⊕2 This easily implies that Xi = P(E), where E = OP1 (1) ⊕ OP1 . The curve R sits in Xi as the image of a section deﬁned by the projection E → OP1 . Its normal ⊕2 N O 1 (−1) sheaf Ri/Xi is isomorphic to P . By adjunction, the anticanonical line bundle of Xi restricted to Ri is isomorphic to 2 ∼ ∼ ORi (2) ⊗ Λ (NRi/Xi ) = ORi (2) ⊗ ORi (−2) = ORi .

Thus −KXi is relatively numerically trivial. This proves that the birational map f is a ﬂop. Let π : Q0 → Q be the resolution of singularities of Q equal to the proper 4 transform of Q under the blow up of the point o in P . Its exceptional divisor E 1 1 0 is isomorphic to P × P . There are two projections Q → Xi, i = 1, 2, which 1 restricts to E as the two projections E → P . n n Example 5.1.2. Let T : P 99K P be the standard Cremona involution. Let X be the blow-up of the vertices of the coordinate simplex. Consider the birational 0 map T : X 99K X obtained by lifting T to X. It is a small birational map that restricts to a birational map of the proper transform of each facet of the simplex to the exceptional divisor over the opposite vertex of the simplex. More generally, for any k > 1, it sends the proper transform of any face of codimension k to the proper transform of the opposite face of codimension n − k − 1. Let us see that T 0 is an example of a ﬂop. We will do it only in the case n = 3 leaving the general case to the reader. Consider the linear system of quadrics in 3 P that passes through the vertices of the coordinate simplex. Its dimension is equal to 5. Its proper transform on X has no base points and contracts the proper transforms Rij of the six edges of the coordinate simplex. It is equal to the linear 3 system |2e0−e1−e2−e3−e4|, where e0 is the pre-image of a plane in P and ei are the divisor classes of the exceptional divisors over the base points. Since −KX = 4e0 − 2e1 − 2e2 − 2e3 − 2e4, we see that −KX is relatively numerically trivial. 6 The image of X is a closed 3-dimensional subvariety Z of P with 6 singular points. Choose a basis in the linear system of quadrics formed by the quadrics 100 LECTURE 5. FACTORIZATION PROBLEM

5 V (t0t1),V (t0t2),...,V (t2t3). It gives the birational morphism φ : X → P which contracts the curves Ri to the points [1, 0,..., 0],... [0,..., 0, 1]. Then the 0 0 5 composition φ : T ◦ φ differs from φ by the projective transformation g of P −1 given by [z0, . . . , z5] 7→ [z5, z4, . . . , z0]. The composition φ ◦ g ◦ φ is equal to the composition of 6 flops. 3 One can give a toric interpretation of each flop. Consider the toric variety P with the standard fan Σst defined by the vectors e1, e2, e3, e0 = −e1 − e2 − e3 3 in R . For any subset J of {0, 1, 2, 3|} denote by FJ the convex cone spanned by the vectors ej, j ∈ J. The interior of FJ corresponds to the orbit OJ of dimension 3 − #J and OJ ⊂ OK if and only if K ⊂ J. More precisely, we can choose 3 coordinates in P such that OJ is the coordinate subspace tj = 0, j ∈ J. Blowing up a vertex of the coordinate simplex OJ , #J = 3, we obtain the toric variety whose fan is generated by its 1-skeleton obtained by adding to the 1-skeleton of P Σst the ray RJ = R≥0( j∈J ej). Let us blow-up two vertices say p3 = O012 and p4 = O013. The new fan Σ acquires two new rays R012 = R≥0(−e3), and R013 = R≥0(−e2). Note that the cone spanned by these two rays does not belong to Σ because its intersection with the cone F01 ∈ Σ is equal to the ray R(e0 + e1) which does not belong to Σ. The proper transform `34 of the edge hp3, p4i corresponds to the cone F01 ∈ 0 Σ. Let Σ be the new fan obtained from Σ by deleting the cone F01. Let φ : XΣ → XΣ0 be the toric morphism corresponding to the map of fans Σ → Σ1 under which F01 is mapped to the cone spanned by R012,R013,F0,F1. The morphism φ is a small contraction of the curve `34 to the 0-dimensional orbit defined by the 0 + non-simplical cone hR012,R013,F0,F1i ∈ Σ . Now let us consider the fan Σ obtained from Σ1 by adding the 2-face spanned by R012 and R013 (now there is no problem because we have deleted the face spanned by F0,F1). We have a toric + morphism φ : XΣ+ → XΣ0 which blows down the orbit corresponding to the face hR012,R013i. This a commutative diagram of rational maps

f XΣ / XΣ+

φ φ+ " | XΣ0

By adjunction formula, deg KP3 · hp3, p4i = −2. This easily implies that KXΣ · 0 `34 = 0. Thus f is a ﬂop. Note that adding to Σ the ray equal to the intersection 00 of the 2-faces hR012,R013i and hF0,F1i, we obtain a fan Σ such that the natural morphism σ : XΣ00 → XΣ0 is a resolution of singularities with exceptional divisor isomorphic to a nonsingular 2-dimensional quadric. There are two projection 00 00 + Σ → Σ and Σ → Σ which deﬁne two new morphisms ν:XΣ00 → XΣ and 5.1. ELEMENTARY LINKS 101

ν+ : XΣ00 → XΣ which make the following diagram commutative:

XΣ00 ν ν+

| f # XΣ / XΣ+

φ φ+ " { XΣ0

+ For future use, observe that the fans Σ and Σ are not isomorphic. The fan Σ+ contains the two opposite faces he2, e3i and h−e2, −e3i. The fan Σ does not have 0 3 such faces. Consider the projection to N → N = Z /Z(−e2). The image of 3 0 Σ+ ⊂ NR = R is a N -fan with 1-skeleton R≥0 equal to the images of the vectors + e1, e3, −e3, −(e1 + e3). It deﬁnes the toric surface F1. The 2-face he2, e3i of Σ is mapped to the 1-face R≥0e3. However, it is easy to see, looking at the extremal rays in the Mori cone, that XΣ does not admit any morphisms onto F1.

Another type of birational transformation we will need is an elementary transformation of projective bundles. Let P(E) → S be the projective bundle associated to a rank r locally free sheaf E over a smooth variety variety S. Let Z be an effective divisor on S considered as a closed subscheme i : Z,→ S and F be a locally free sheaf on Z of rank 0 r < r. Suppose we have a surjection u : E → i∗F. It deﬁnes a closed embedding su : P(F) ,→ P(E). When F is of rank 1, this is a section over Z. 0 Since the projective dimension of OZ is equal to 1, the sheaf E = Ker(u). ∨ is a locally free sheaf on S of rank r. The projective bundle P(E )2. is denoted by elm(P(E), u, F) and is called the projective bundle obtained by an elementary transformation along the data (F, u). Note that

0 c(E ) = c(E)/c(i∗F). (5.1)

Since E0 and E are isomorphic outside Z, the two projective bundles are isomor- 0 phic over the complement of Z. When Z is smooth, one can show that P(E ) is obtained from P(E) by blowing up su(Z), followed by the blowing down the proper transform of the pre-image of Z in P(E). 1 Example 5.1.3. Let S = P and X = Fn = P(OP1 ⊕ OP1 (−n)) Take a closed 1 point y ∈ P and F = Oy the structure sheaf of y.

2 • We use Grothendieck’s notation P(E) := ProjS E 102 LECTURE 5. FACTORIZATION PROBLEM

1 Assume n > 0. Let sn : P → X be the section defined by the surjection OP1 ⊕ OP1 (−n) → OP1 (−n). The image of sn is the unique curve on X with negative self-intersection equal to −n. It is called the exceptional section. Consider 0 a surjective map of sheaves u : OP1 ⊕ OP1 (−n) → Oy and let E be its kernel. 1 We know that every locally free sheaf over P is isomorphic to a sheaf OP1 (−a) ⊕ OP1 (−b). Computing the degrees, we find that a + b = n + 1. If both a, b < n, we 0 get E is contained in the summand OP1 , so the cokernel of u is not a sky-scrapper sheaf. So, up to switching a, b, we have only two possibilities (a, b) = (0, n + 1) or (a, b) = (1, n). In the former case, u induces an isomorphism on H0, and hence u factors through the projection to OP1 (−n). This shows that u defines a point 1 on the exceptional section sn(P ). The elementary transformation produces the ∼ surface Fn+1 = P. In the latter case u factors through OP1 . Then u defines a point outside of the exceptional section. The elementary transformation produces the surface Fn−1. 0 ∼ Assume n = 0. Then, by similar argument, we obtain that E = OP1 ⊕ ∼ OP1 (−1), hence the elementary transformation produces the surface F1 = P(OP1 ⊕ ∼ OP1 (−1)) = P(OP1 ⊕ OP1 (1)).

There are 4 types of elementary links between Mori ﬁbre spaces X → S and X0 → S0. Type I: ﬂaps Z / X0

X S0

S ~

Here Z → X is a divisorial contraction of an extremal ray R with KZ · R < 0, and S0 → S is a morphism with relative Picard number equal to 1.

Type II: ﬂaps Z / Z0

X X0

~ S S0 5.1. ELEMENTARY LINKS 103

Here Z → X and Z0 → X are extremal divisorial contractions as in the previous case.

Type III: ﬂaps X / Z

S X0

~ S0 This link is the inverse of a link of type I.

Type IV: ﬂaps X / X0

S S0

~ T Here the morphisms S → T and S0 → T are morphisms to a normal variety T with relative Picard numbers equal to 1. Example 5.1.4. Let us specialize to dimension 2. There are no flips here, and all varieties are nonsingular. In type I, S could be a one-point variety. In this case X must a Del Pezzo 2 surface with Picard number 1, i.e. S = P . The morphism Z → X is the blow-up 0 1 0 ∼ 0 1 of one point. Then S = P and Z = X = F1 → S is the P -bundle structure on F1. 1 1 1 In type II, S = P , X → P is a P -bundle, Z → X blows up one point, and Z → X0 blows down the proper transform of the fibre through this point. So, X0 → 1 0 P is another projective bundle, and X 99K X is an elementary transformation. 0 1 1 Type III is the inverse of type I. In Type IV, X = X = P × P , T is the 1 one-point varietry, and the two Mori fibrations are the two projections to P . 2 2 Example 5.1.5. Consider the standard Cremona transformation T : P 99K P . 2 We start with the Mori fibre space X = P → point. Then we blow up the point p1 = [1, 0, 0] to get a morphism Z → X. The projection from the point p1 defines 1 1 a P -bundle structure Z → P . This gives us a first link between X → point and 0 1 X = Z → P . It is of type I. Next we make an elementary transformation at 104 LECTURE 5. FACTORIZATION PROBLEM the point p2 = [0, 1, 0]. This is a link of type II leading to one of the projections 1 1 1 F0 = P ×P → P . Next we make an elementary transformation at the pre-image 1 1 of the third base point [0, 0, 1] on P × P . This is a link of type II. We obtain the 1 minimal ruled surface F1 → P . Finally, we blow down the exceptional section 2 F1 → P . This is a link of type III.

Z0 Z00

~ } ! Z F1 F0 F1 Z

II II III 2 1 1 1 2 P I P P P P

~ pt pt

We leave to the reader the job of doing the same for degenerate standard Cremona transformations with two or one base points.

3 3 Example 5.1.6. Consider the standard Cremona transformation T : P 99K P . We 3 start with the Mori fibre space X = P → point. Next we blow up the first point 1 2 p1 = [1, 0, 0, 0]. The projection from this point defines a P -bundle X1 → P . This is our first link. Next we blow-up the pre-image of the second base point p2 = [0, 1, 0, 0] and make a flop along the proper transform of the edge of the coordinate simplex joining the points p1, p2. It follows from Example 5.1.2 that the new surface admits a morphism to F1. This is our new Mori fibre space. Then we blow up the the third vertex p3, make two flops along the edges hp1, p3i, hp2, p3i and get a morphism to D7, the blow-up of 2 points (a Del Pezzo surface of degree 7). Next we blow up the vertex p4, make three flops along the edges hpi, p4i, i = 1, 2, 3 and get a morphism to X3 → D6, a Del Pezzo surface of degree 6. The 3-fold X3 is a toric variety, the corresponding fan contains 8 rays spanned by the vectors ±ei, i = 0, 1, 2, 3. It has 18 two-dimensional cones h−ei, −eji, h−ei, eji, i 6= j. 3 It is isomorphic to the blow-up of 4 points in P . We can start blowing down the four exceptional divisors making square morphisms of Mori fibre spaces. 5.1. ELEMENTARY LINKS 105

Z1 Z2 Z3

flop 2 flops 3 flops

3 ~ ~ ~ 3 P o X1 X2 X3 X4 / X5 / X6 / X7 / P

I II II II II II II I 2 2 pt o P o F1 o D7 o D6 / D7 / F1 / P / pt n Example 5.1.7. Consider a non-degenerate quadratic transformation in P , n > 2. Recall that it is decomposed as the blow-up of a nonsingular quadric Q0 in a hyperplane, followed by the projection from a point on the image quadric Q ⊂ n+1 P that does not lie in the proper transform of the hyperplane containing the quadric. Its decomposition into elementary links is as follows.

Z Z0

n ~ n P II Q II P

~ pt pt Note that a nonsingular quadric of dimension > 2 is an example of a Fano variety with Picard number equal to 1.

Example 5.1.8. Let X = (O⊕3) =∼ 1 × 2 → 1 be the trivial 2-bundle. P P1 P P P P 1 ⊕3 Take a point y ∈ and a surjection O 1 → Oy. The kernel is isomorphic to E = P P ⊕2 0∨ ⊕2 O 1 ⊕OP1 (−1). The corresponding projective bundle P(E ) = P(O 1 ⊕OP1 (1)) P 3 1 P is isomorphic to the blow-up of a line in P with the morphism to P defined by the 3 projection from the line. Let a1 : P(E) → P be the birational morphism which is 1 2 the inverse of the blow-up of the line. Let b1 : P × P 99K P(E) be the birational map defined by the elementary transformation. 1 2 ⊕2 2 Consider the other projection X = P × P = P(OP2 ) → P and make ⊕2 the elementary transformation with respect to the surjection O 2 → O`, where ` P is a line. The elementary transformation produces the projective bundle P(OP2 ⊕ 3 2 OP2 (1)). It is isomorphic to the blow-up a point in P with the morphism to P defined by the projection from the point. Let a2, b2 be the similarly defined rational maps. Thus we get a chain of elementary links decomposing the birational map T : 3 3 −1 −1 P 99K P equal to the composition (a2 ◦ b2) ◦ (b1 ◦ a1 ). 106 LECTURE 5. FACTORIZATION PROBLEM

Z0 Z

$ 3 a1 b1 1 2 2 1 b2 $ 0 a2 3 P oo P(E) o P × P P × P / P(E ) // P

III II IV II III 1 1 2 2 pt o P P P P / pt

# pt {

The map T can be described as follows. Fix a point xi and a line ì not contain- 2 1 ing xi and identify P with the family of lines through xi and P with the family of planes through ì. The line ` corresponds to the point y over which we do the −1 elementary transformation. The point x1 is the base point of a1 . 3 Fix a general plane Π1 and a general line ` in P . Identify Π with the family of lines through xi by taking the intersection of a line hx, xii with the plane Π. 1 Identify P with the family of planes through ì by taking the intersection of a plane hx, ìi with `. Now consider the following map. Take a general point x ∈ 3 P . The line `x = hx, x1i and the plane Πx = hx, Π1i defines a line through x2 and the plane through `2. We take T (x) to be the intersection point of this line and this plane. Let us see the indeterminacy points of T . Let Ai = ì ∩ Π and Bi = hxi, ìi ∩ `. Obviously, the image T (x) is not defined if x = x1 or x ∈ `1. 0 Also, T is not defined at any point x in the intersection `1 of two planes h`1,B2i and hx1,A2,B2i. Thus we see that the base locus contains x1 and the union of 0 two intersecting lines `1 + `1. It is easy to check that the pre-image of a general 3 plane in the target P under the map a2 ◦ b2 is a divisor of type (1, 1). The map −1 −1 3 2 1 b1 ◦a1 : P 99K P ×P is given by the 5-dimensional linear system of quadrics 3 5 2 1 through x1 and `1. It maps P to P with the image equal to P × P embedded by the Segre map. This shows that T −1(plane) is a quadric, thus T is a degenerate quadratic transformation. Example 5.1.9. Consider the Cremona transformation defined by the homaloidal linear system of cubics through the double structure of a rational normal curve C 3 in P . Let |L| = |2H − C| be the 2-dimensional linear system of quadrics through 1 3 2 C. It defines a structure of a P -bundle on the blow-up q : X1 = BlC P → P . 2 ∨ −1 For any point x ∈ P = |L| , the fibre q (x) is the secant line `x contained in the base locus of the pencil of quadrics in |L| defined by this point. The proper transform F of the tangent surface Tan(C) is mapped to the conic K in the plane 5.1. ELEMENTARY LINKS 107 parameterizing the tangent lines of C. The exceptional divisor E1 is a finite cover 2 of P of degree 2. The two surfaces are tangent along the curve R which is a section of the bundle over the conic K (recall that Tan(C) contains the curve C as its cuspidal curve). The curve R is the ramification curve of the double cover 2 q|E1 : E1 → P .

Let ν : Z = BlRX1 → X1 be the blow-up of R. Computations from section ⊕2 2.28 show that the normal bundle of R in X1 is isomorphic to OP1 (3) . This shows that the exceptional divisor E2 of ν is isomorphic to F0. The proper trans- 0 0 form F of F in Z intersects E2 along the curve R of type (1, 1). The proper 0 transform of E1 intersects E2 along the same curve. The union of E2 and F is the 0 2 pre-image of the conic K under the projection q ◦ ν : Z → X1 → P . Now we 0 0 2 blow down F to a curve R on the new projective bundle X2 → P . In other words 2 we make an elementary transformation of X1 → P along the curve R given by an invertible bundle L on K of degree 1. 2 A fibre of q : X1 → P is naturally identified with a secant ha, bi of C and 2 (2) the plane P with the symmetric square C . The map {a, b} → ha, bi defines an 3 embedding of C into the Grassmannian of lines in P . The image of the embedding in the Plucker¨ space is a Veronese surface V ⊂ G(2, 4). The morphism q becomes ∼ isomorphic to the universal family of lines over the Veronese surface. Thus X1 = P(E), where E is isomorphic to the pre-image of the universal subbundle over 3 G(2, 4) restricted to V . Since a general point in P lies on a unique secant and any general plane contains three secants, we obtain that the cohomology class of V is of type (1, 3). his allows us to compute the Chern classes of E. We have c1(E) = 0 2, c2(E) = 3. The first equality implies that E is self-dual. The Chern classes of L are c1(L) = 2, c2(L) = 1. Applying (5.1), we obtain that the new vector 0 ∼ 0 0 bundle E = obtained by the elementary transformation has c1(E ) = 0, c2(E ) = 0 ∼ ∼ 0 2. One can show that E = E(1), so P(E) = P(E ). After making the elementary 3 transformation, we define the blowing down X2 → P with the exceptional divisor 0 equal to the image of E1.

3 ~ 3 P o X1 II X2 / P

I I 2 2 pt o P P / pt 108 LECTURE 5. FACTORIZATION PROBLEM

5.2 Noether-Fano-Iskovskikh inequality

Let HX be a linear system without ﬁxed components on a smooth variety X which deﬁnes a rational map X 99K Y to a normal variety Y , so that HX is equal to the proper transform of an ample linear system HY on Y . We will use a log resolution σ : X0 → X of its base scheme. Recall that it is given by a sequence of birational morphisms

0 σN σN−1 σ2 σ1 X = XN −→ XN−1 −→ ... −→ X1 −→ X1 = X, where each morphism σi : Xi+1 → Xi is the blowing up of a smooth closed subscheme Bi of Xi, which we can always assume to be of codimension ≥ 2. For any N ≥ a > b ≥ 1, we set

σab = σa ◦ ... ◦ σb.

We will often identify Bi with a closed subvariety of Xj, j ≤ i, if the morphism σij : Xi → Xj is an isomorphism at the generic point of Bi. −1 Let Hi = σi1 (H) be the proper transform of H in Xi. Then Bi is contained in its base scheme and we set

mi = min multηi D, D∈Hi where ηi is the generic point of Bi. It follows from Hironaka’s Theorem on resolution of singularities that we can also assume that the following normal ﬂatness condition is satisﬁed

• mi > 0, mi ≥ mj, 1 ≤ i ≤ j ≤ N.

−1 • A general divisor of the linear system σi1 (H), the scheme-theoretical inter- −1 −1 section of two general divisors in σi1 (H), and the base scheme of σi1 (H) are normally ﬂat along Bi (in particular, it is equimultiple along Bi).

Let E¯i be the exceptional divisor of σi : Xi+1 → Xi. It is the projective bundle over B defined by the normal bundle (N ∨ ) of B in X . It is a reduced i P Bi/Xi i i effective divisor in Xi. We denote by Ei its proper transform in X and by Ei its scheme theoretical pre-image in X. We define a partial order on the set of subvarieties Bi by writing Bi > Bj if σij(E¯i) ⊂ E¯j. The Enriques diagram of the resolution is an oriented graph whose vertices are the subvarieties Bi and an edge connects Bi with Bj if Bi > Bj and Bi is contained in the proper transform of E¯j in E¯i under the map σij. If additionally, σi,j : E¯i → Xj maps Bi isomorphically onto Bj, we say that Bi is infinitely near 5.2. NOETHER-FANO-ISKOVSKIKH INEQUALITY 109 to Bj of order 1. By induction, we define Bi to be infinitely near of order k to Bj and write Bi k Bj in this case. For any i ≤ j let γij be the sum of the lengths of the paths connecting i and j in the Enriques diagram. In the case of surfaces, all subschemes Bi’s are closed points xi ∈ Xi−1 and ¯ 1 the exceptional divisors Ei are isomorphic to P . In this case the notion of being infinitely near makes the partial order on the set of points Bi’s. Two points xi, xj are connected by an edge (xi, xj) if and only if xi 1 xj. The standard formula for the behavior of the canonical class under the blow-up of a smooth closed subschemes shows that

N N ∗ X ∗ X X KX = σ (KX0 ) + δiEi = σ (KX0 ) + ( γijδj)Ei, (5.2) i=1 j=1 j≤i where δj = codim (Bi,Xi−1) − 1. −1 Let Hi = σi1 (H) be the proper transform of the linear system H to Xi. It is ∗ ¯ equal to σi1(Hi−1) − miEi. By induction, we obtain

N N −1 ∗ X ∗ X X σ (H) = σ (H) − miEi = σ (H) − ( γijmj)Ei, (5.3) i=1 i=1 j≤i

P n For example, if H = |dH − miZi| is a smooth homaloidal linear system in P , we may take for a resolution the sequence of blow-ups of Bi = (Zi)red. In this case the Enriques diagram consists of disjoint vertices, and we obtain

∗ X KX = −(n + 1)σ (H) + aiEi, −1 ∗ X σ (H) = dσ (H) − miEi.

Definition 5.2.1. Let H be a linear system without fixed components on a variety X. We say that the pair (X, bH), where 0 ≤ b ≤ 1 is a rational number, has canonical singularities (resp. terminal) if there exists a log resolution of Bs(H) with δi − bmi ≥ 0, i = 1,...,N, resp. > 0. The canonical threshold (terminal threshold of H is maximal positive rational number c(H) such that the pairs (X, cH) has canonical (terminal) singularities. An irreducible component Ei is called crepant if c(H) = ai/ri. One can show that these definitions do not depend on a choice of a log resolution and coincide with the definition of the canonical (terminal) pair (X, bD), where D is a general divisor in H. 110 LECTURE 5. FACTORIZATION PROBLEM

Thus H has canonical singularities if δi ≥ bmi ≥ 0 for all i. Also, we see that δ c(H) = min{ i } (5.4) mi The following theorem is called the Noetehr-Fano-Iskovskikh inequality.

Theorem 5.2.1. Let f : X 99K Y be a birational map deﬁned by a linear system −1 HX = f (HY ) for some ample linear system HY on Y . Assume that, for some rational number t, we have |tHY + KY | = ∅ but |tHX + KX | 6= ∅. Then t < c(HX ).

Proof. Let (X0, π, σ) be a log resolution of f. We have

N −1 ∗ X ∗ π (HX ) = π (HX ) − miEi = σ (HY ). i=1

∗ P ∗ P Write KX0 = π (KX ) + δiEi, multiply both sides by t, add π (KX ) + aiEi, to the left-hand side and KX0 to the right side to obtain

N ∗ ∗ X KX0 + σ (tHY ) = π (KX + tHX ) + (δi − tmi)Ei. (5.5) i=1

Assume δi ≥ tmi ≥ 0 for all i. Applying σ∗ to both sides and using that σ∗(KX0 ) = KY , we get

N ∗ X ∅ = |KY + tHY | = |σ∗(π (KX + tHX ) + (δi − tmi)σ∗(Ei)|. i=1

Since the right-hand side is non-empty, we get a contradiction. Thus δi − tmi < 0 for some i, hence t > c(H).

Let α : X → S, and α00 : Y → S0 be Mori ﬁbrations. Since the relative Picard numbers are equal to 1, we can write

∗ 0 0∗ 0 HX ⊂ | − rKX + α (A)|, HX = −r KX + α (A ) for some rational positive numbers r, r0 and some effective divisor classes on S 0 0 0 and S . Then tHY + KY | = ∅ for t > r . So if r > r , then |tHX + KX | 6= ∅ for r ≥ t > r0. Thus c(H) < r0 (5.6) 5.2. NOETHER-FANO-ISKOVSKIKH INEQUALITY 111

Together with (5.4), this implies that

mi 1 max{ } > 0 . δi r

n Let us apply this to the case when X = Y = P , HX ⊂ |dh| is a homaloidal n linear system on P and HY = |h|, where h = c1(OPn (1)). Since KPn = −(n + 0 n+1 1)h, we obtain that r = d , hence n + 1 c(H ) < , (5.7) X d and m d max{ i } > . δi n + 1 For example, if n = 2, we obtain that there exist a base point of multiplicity > d/3. This of course follows from Noether’s inequality

m1 + m2 + m3 > d, if m1 ≥ m2 ≥ ... ≥ mN . PN P Let us rewrite i=1(δi −tmi)Ei as the sum j≤i γij(δj −tmj)Ei. If all these sums are non-negative, then t ≤ c(HX ). Thus P n j≤i γijδj o c(H) = min P . i,j j≤i γijmj

Hence, in view of (5.6), we obtain P n j≤i γijmj o 1 max P > 0 . (5.8) i,j j≤i γijδj r

Deﬁnition 5.2.2. The exceptional divisor Ei is called the maximal singularity of the linear system H if it satisﬁes (5.8).

Assume that all δj ≥ δi for any i ≤ j (e.g. dim X = 2). Let Ej0 corresponds to the minimal vertex in the Enriques diagram, i.e. σj01 : Xj → X is an isomorphism at the generic point of Bj. Then, by the choice of a resolution, we have mj0 ≥ mj for any j ≤ i with γij 6= 0. This implies that

P γ m P γ m mj0 j≤i ij j0 j≤i ij j 1 = P ≥ P > 0 . δj0 δj0 j≤i γij j≤i γijδj r 112 LECTURE 5. FACTORIZATION PROBLEM

So, we can find the maximal singularity among the irreducible components of the base scheme and its multiplicity m satisfies δ m ≥ , r0 where δ + 1 is equal to the codimension of the component. Example 5.2.1. Let us look at Example 2.5.3. The Enriques diagram consists of 4 minimal vertices corresponding to the for points in the base locus, and the fifth vertex corresponding to a line infinitely neat to the fourth vertex. All the multiplicities mi are equal to 1. Any of the minimal vertices give mi/δi = 1/2. we have r0 = 2/4 = 1/2. So, none of them is maximal. For the fifth vertex the ratio (5.8) is equal to 2/3, and this is larger than 1/2. So, the component E5 of the exceptional divisor of the resolution is maximal. 0 For any morphism φ : X → X we denote by ≡φ-equivalence of Q-Weil divisors (where D ≡φ 0 if D · C = 0 for any curve C with σ(C) = point). The following is known as the Negaivity Lemma. We refer for the proof to [Kollar]. Lemma 5.2.2. Let σ : X0 → X be a birational morphism with exceptional irreducible divisors Ei.A Q-Weil effective divisor D is called σ-effective if no Ei appears in D. Assume that X aiEi ≡φ H + D for some σ-effective divisor D and σ-nef Q-Cartier divisor H. Then all ai are non- −1 positive. Moreover, if σ(Ei) = x ∈ X and the restriction of D or H to σ (x) is not numerically trivial, then ai < 0. 0 Corollary 5.2.3. Let X,Y be Q-factorial varieties, and f : X 99K Y be a small birational map. Let H be an ample divisor on X such that H0 = f(H) is nef. Then f −1 is a morphism, hence f is an isomorphism if H0 is ample. Proof. Choose a smooth resolution (X, π, σ). It suffices to show that, for any point y ∈ Y , dim π(σ−1(y)) = 0. Since f is small, the exceptional divisors ∗ 0 Ei of π are the same as exceptional divisor of σ. Then σ∗(π (H)) = H and ∗ 0 0 σ∗(σ (H )) = H implies

X ∗ 0 ∗ aiEi = σ (H ) − π (H) = (π − nef) + (π − effective).

Also

X ∗ ∗ 0 − aiEi = −π (H) + σ (H ) = (σ − nef) + (σ − effective). 5.2. NOETHER-FANO-ISKOVSKIKH INEQUALITY 113

∗ 0 ∗ By the Negativity Lemma, we get that all ai = 0. Thus σ (H ) = π (H). Suppose dim π(σ−1(y)) > 0 for some point y. Take a curve C in σ−1(y) such that π(C) is not a point. Then

0 ∗ 0 ∗ 0 = H · σ∗(C) = σ (H ) · C = π (H) · C = H · π∗(C) > 0.

This is a contradiction.

Note that in the case of surfaces, this is well-known and follows from negative definiteness if the intersection matrix of exceptional divisors. Theorem 5.2.4. Let f :(φ : X → S) → (φ0 : X → S0) be a birational isomorphism of Mori fibre space defined by a linear system without fixed components ∗ H ⊂ |D|. Write D in the form −µKX + φ (A) for some effective divisor class A on S and a rational positive number µ (this is possible, by definition of a Mori −1 0 0 0 0 0 0 fibre space). Let H = f (H ), where H ⊂ |D | = | − µ KX0 + φ (A )| for some ample D0 on X0 and ample A0 on S. Then (i) µ ≥ µ0 and µ = µ0 if and only if f is square;

1 1 (ii) if (X, µ HX ) is a canonical pair and |KX + µ HX | is nef, then f is a Sarkisov isomorphism.

∗ P Proof. Let us consider equality (5.5). Substitute KX0 = σ (KY ) + biFi for some exceptional divisors Fi of σ. Then we get

∗ X ∗ X π (KX + tHX ) + (ai − tri)Ei = σ (KY + tHY ) + biFi. (5.9)

Let Γ be a general curve contained in ﬁbres of X → S which is disjoint from φ∗(A) and all π(Ei)’s. The latter condition allows one to identify Γ with its pre-image in X. Intersecting both sides with Γ, we obtain

∗ ∗ X π (KX + tHX ) · Γ = (σ (KY + tHY ) · Γ + biFi · Γ.

Take t = 1/µ0, then the right-hand side is equal to 1 X σ∗(φ0∗(A0)) · Γ + b F · Γ. µ0 i i

0 Since X has terminal singularities all bi’s are positive. Also A is ample. This implies that the right-hand side is non-negative. Since the left-hand side is equal µ µ 0 0 to (1 − µ0 )KX · Γ, we get 1 − µ0 ≤ 0, i.e. µ ≤ µ. Moreover, if µ = µ , then 0 0 ∗ 0 φ (A ) ··σ∗(Γ) = 0, thus f(Γ) = σ∗(π (Γ)) is contained in ﬁbres of φ . This means that f is square. This checks (i). 114 LECTURE 5. FACTORIZATION PROBLEM

Assume that (HX , µ) is canonical. Then, if we take t = 1/µ in (5.9), we obtain 0 ri P ri that the coefficients a = ai − µ in the sum (ai − µ )Ei are all non-negative. Replacing now Γ with a general curve Γ0 contained in fibres of φ0, and repeating the previous argument, we obtain µ ≤ µ0. By (i), f is square. Let Ei = Fj for some subset K of indices i ∈ I. Consider the equality 1 X 1 X π∗(K + H ) + a0 E = σ∗(K + H ) + b F X µ X i i Y µ Y i i It gives X X 1 a0 E − b F ≡ σ∗(K + H ), i i i i π Y µ Y X X 1 b F − a0 E ≡ π∗(K + H ). i i i i σ X µ X 0 If Ei = Fj for some i and j, then Ei enters in the equality with coefficients ai − bj and, in the second equality, with the coefficient bj − ai. The right-hand sides are the sums of a relative nef and relative effective divisors. Applying the Negativity Lemma, we get that all non-zero coefficients and both equalities are non-positive, 0 in particular ai = bj if Ei = Fj. So, we obtain that each exceptional irreducible divisor of σ is equal to a non-crepant exceptional irreducible divisor of π and all 0 other exceptional Ei are crepant. Now let ν : Z → X be the extraction of the crepant exceptional divisors of π 0 and let f = f ◦ ν : Z 99K Y be the composition of the rational maps. Then the exceptional irreducible divisors of f and f −1 are the same. In particular, f 0 is a small birational map. Let us show that f 0 is an isomorphism. Choose a very ample divisor on Z and D0 = f 0(D) be its image in Y . Since 1 K + H + δD = (φ ◦ ν)∗(A) + δD Z µ Z is ample (and terminal) for 0 < δ << 1 and similarly 1 K + H + δD = (φ ◦ ν)∗(A) + δD Y µ Y Then σ has the same set of irreducible components and we have 1 π∗(K + H ) + δD0 = (φ ◦ ν)∗(A0) + δD0 X µ X is ample (and terminal) for 0 < δ << 1. Applying Lemma ??, we obtain that f 0 is an isomorphism. In particular, g = f −1 is a crepant morphism (i.e. all of its exceptional divisors are crepant). 5.3. THE UNTWISTING ALGORITHM 115

We have 1 1 1 g∗(K + H ) = K + H = φ0∗( A0). X µ X Y µ Y µ This implies that g factors through φ0 : Y → S0, hence g, and hence f, is an isomorphism. Since we know that f sends ﬁbres of φ to ﬁbres of φ, f induces an isomorphism S =∼ S0.

5.3 The untwisting algorithm

In this section we explain how the Sarkisov program works. The main idea (called φ a 2-ray game) is as follows. We start with a birational map f :(X → S) → φ0 (X0 → S0) between two Mori fibre spaces. It is given by a linear system H on X −1 without fixed components such that HX = f (HX0 ), where HX0 is a very ample 0 ∗ 0 0∗ 0 linear system on X . We write HX = −µKX + φ (A), HX0 = µ KX0 + φ (A ). We introduce the invariants, called the Sarkisov degree. They form a triple ∗ (µ, c, e) which consists of the number µ such that H = −µKX + φ (A), the canonical discrepancy c(H) and the number e of crepant divisorial valuations for H. We order lexicographically the Sarkisov degrees. Suppose f is not an isomorphism of Mori fibre spaces. By Theorem 5.2.4, there are two cases to consider

1 (i) (X,KX + µ H) does not have canonical singularities;

1 1 (ii) (X,KX + µ H) has canonical singularities but KX + µ H is not nef. 1 If the first case occurs, then c < µ . We apply Proposition 5.3.1 below to find ∗ a blow-up ν : Z → X with ρZ/X = 1 such that KZ + cHZ = f (KX + cHX ), −1 where HZ = σ (HX ). It is called a maximal extraction. In the case of surfaces, this is just the blow-up a base point with maximal multiplicity. One shows that a maximal extraction strongly decreases the Sarkisov degree. Since ρ(Z/S) = 2, there will be two relative extremal rays in NE(Z). One of them P defines a relative contraction (Z/S) → (X/S). We look at another one Q and consider its contraction (Z/S) → (Y/S). If Y is in the Mori category, we obtain a new Mori fibre space Y/S. This is our first link of type I or II. If Y/S is not a Mori fibre space, we make a small contraction (Z0/S) → (Y/S) with flapping ray P 0. Again the relative Picard number of (Z0/S) is equal to 2, so we can find another extremal ray Q0. We repeat the game hoping that a sequence of flaps terminates (it does in dimension 3). In case (ii) we make the link by choosing some contraction S → T as follows. 1 Let P ∈ NE(X) be the ray defining the Mori fibration X → S. Since KX + µ H is 116 LECTURE 5. FACTORIZATION PROBLEM

1 1 not nef, we ﬁnd an extremal ray Q with (KX + µ HX )·Q < 0. Since KX + µ H = φ∗(A), the rays P and Q are different. Then we show that the extremal face hP,Qi exists, so we can contract this face to get X → Y → T equal to X → S → T . This step decreases the Sarkisov degree. 0 1 The process ends when the new µ becomes equal to µ and new K + µ H becomes canonical and nef. Then we obtain a Sarkisov isomorphism.

Proposition 5.3.1. Let H be a linear system without ﬁxed components deﬁning a birational map f : X 99K Y . There exists Z with at most Q-factorial terminal singularities and a Mori extremal contraction ν : Z → X such that (Z, c(ν−1(H))) has canonical singularities and

−1 ∗ KZ + cν (H) = ν (KX ) + cH).

Here c is the canonical threshold of H. The morphism c is called the maximal extraction.

Proof. If all dimensions of the base schemes Bi in a resolution satisfy dim Bi ≤ dim Bj if Bi > Bi, this is easy. We choose a maximal singularity represented by an irreducible component of the base scheme, and blow it up. The corresponding birational morphism ν : Z → X is our maximal extraction. Let us consider the general case. Consider a log resolution π : Z → X. Let c be the canonical threshold of (X, H). Let t0 be minimal t ≥ 0 such that

∗ X KZ + t0HZ = π (KX + t0HX ) + (ai − t0ri)Ei is nef. Consider an extremal ray R in NE(Z/X) such that KZ · R < 0 and (KZ + t0HZ ) · R = 0. We ﬁnd such R represented by a curve in some Ei such that ai − t0ri = 0

Let Z → Z1 be the contraction of R and HZ1 be the image of HZ . We deﬁne t1 ≥ t0 as above, and continue by induction until we deﬁne (Zi, HZi , ti). The process terminates when we are left with only one contraction Zk → X with c = tk. This is our maximal extraction.

5.4 Noether Theorem

Let us see how the Sarkisov Program allows to prove Noether’s Theorem on fac- 2 2 torizations of a Cremona transformation f : P 99K P . 5.4. NOETHER THEOREM 117

Proposition 5.4.1. Let f : Q 99K Q be a birational automorphism of a nonsingu- ∼ 3 lar quadric Q = F0 in P . Consider it as a Mori ﬁbre space X0 → S by choosing 1 one of the rulings φ1 : Q → S = P . Then f is a composition of links of type 1 II and IV between Mori ﬁbre spaces Xi → P , where Xi = F0 for even i and Xi = F1 for odd i.

Proof. Let H be a linear system on X0 = Q defining a birational map f : Q 99K Q. 0 We choose the very ample linear system H = OQ(1) on the target quadric. The linear system H is contained in the complete linear system |af1 +bf2|, where f1, f2 are the standard generators of Pic(Q). Obviously, we may assume that a ≥ b > 0. We have −KQ = 2f1 + 2f2. Let us consider the Mori fibration on Q defined by 1 the projection p1 : Q → P with fibre of the divisor class f1 so that f2 generates b b the relative Picard group. We have H0 = − 2 KX0 + a − bf1, so that µ = 2 . Obviously, µ0 = 1/2. Applying Theorem 5.2.4, we obtain b > 1, unless the map 0 is an isomorphism, in which case the assertion is obvious. Since |KQ + H | = ∅ 1 but |KQ + µ H|= 6 ∅, we obtain that the canonical threshold c of (Q, H) satisfies 1 2 c > µ = b . Thus we can find a base point x1 of maximal multiplicity r1 > b/2. Let ν1 : Z → Q be the blow-up of x1 with exceptional divisor E1. Let F1 be 1 the proper transform of the fibre of p1 : Q → P passing through x1. We have ∗ ν1 (f1) = E1 + F1. Thus we can write ∗ ∗ ∗ −KZ = 2ν1 (f2) + 2ν1 (f1) − E1 = 2ν1 (f2) + 2F1 + E1, −1 ∗ ∗ ∗ H1 = ν1 (H) = aν1 (f2) + bν1 (f1) − r1E1 = aν1 (f2) + bF1 + (b − r1)E1. ∼ Let σ1 : Z → X1 = F1 be the blowing down of F1. Then

∗ −KX1 = σ1(E1) + 2σ1(ν1 (f2)) = g + 2s, 1 where g = [σ1(E1)] is the class of a ﬁbre of the projection p : F1 → P and s is a section from the divisor class g + s0, where s0 is the exceptional section. We also have a a H = (σ ) (H ) = as + (b − r )g = − K + (b − − r )g. 1 1 ∗ 1 1 2 X1 2 1 0 0 Since b ≥ 2, we have a ≥ 2, hence 2/a ≤ µ and the canonical threshold c of H1 is greater than 1. This gives a maximal base point x2 of multiplicity > a/2. Since s0 ·H1 = b − r1 < b/2 < r1, the point x2 cannot lie on the exceptional section s0. Let ν1 : Z1 → X1 be the blow-up of x2 and then σ1 : Z1 → X2 be the blow-down 1 of the proper transform of the ﬁbre of p : X1 → P through the point x2. The surface X2 is isomorphic to F0 and the birational transformation

−1 −1 tx1,x2 := (σ1 ◦ ν1 ) ◦ (σ ◦ ν ): Q 99K X2 118 LECTURE 5. FACTORIZATION PROBLEM

is equal to the composition of the two elementary transformations elmx2 ◦ elmx1 on Q, where we identify x1 and x2 with their images on Q. Note that the point x2 could be inﬁnitely near to x1.

2 Let us fix a point x0 ∈ Q. The linear projection px0 : Q \{x0} → P defines a birational map. Let l1, l2 be two lines on Q passing through x0 and q1, q2 be −1 their projections. The inverse map px0 blows up the points q1, q2 and blows down 2 the proper transform of the line q1, q2. For any birational automorphism T of P −1 the composition px0 ◦ T ◦ px0 is a birational transformation of Q. This defines an isomorphism of groups

=∼ −1 Φx0 : Bir(Q) → Bir(Q), σ 7→ px0 ◦ σ ◦ px0 .

−1 3 Explicitly, Φx0 is given as follows. Choose coordinates in P such that Q = −1 V (z0z3 − z1z2) and x0 = [0, 0, 0, 1]. The inverse map px0 can be given by the formulas 2 [t0, t1, t2] 7→ [t0, t0t1, t0t2, t1t2].

If T is given by the polynomials f0, f1, f2, then Φx0 (T ) is given by the formula

0 2 0 0 0 0 0 0 [z0, z1, z2, z3] 7→ [f0(z ) , f0(z )f1(z ), f0(z )f2(z ), f1(z )f2(z )], (5.10)

0 where fi(z ) = fi(z0, z1, z2).

Remark 5.4.1. Let z1, . . . , zn ∈ Q be base points of T different from x0. Let −1 −1 T (x0) be a point if T is defined at x0 or the principal curve of T corresponding to x0 with x0 deleted if it contains it. The Cremona transformation Φx0 (T ) is −1 defined outside the set q1, q2, px0 (z1), . . . , px0 (zn), px0 (T (x0)). Here, we also include the case of infinitely near fundamental points of T . If some of zi’s lie on a line li or infinitely near to points on li, their image under px0 is considered to be an infinitely near point to qi. Let Aut(Q) ⊂ Bir(Q) be the subgroup of biregular automorphisms of Q. It o acts naturally on Pic(Q) = Zf +Zg, where f = [l1], g = [l2]. The kernel Aut(Q) 1 1 ∼ of this action is isomorphic to Aut(P ) × Aut(P ) = PGL(2) × PGL(2). The quotient group is of order 2, and its nontrivial coset can be represented by the 1 1 automorphism τ of Aut(P × P ) defined by (a, b) 7→ (b, a).

o Proposition 5.4.2. Let σ ∈ Aut(Q) . If σ(x0) 6= x0, then Φx0 (σ) is a quadratic −1 transformation with fundamental points q1, q2, px0 (σ (x0)). If σ(x0) = x0, then

Φx0 (σ) is a projective transformation. 5.4. NOETHER THEOREM 119

Proof. It follows from Remark 5.4.1 that Φx0 (σ) has at most 3 fundamental points if σ(x0) 6= x0 and at most 2 fundamental points if σ(x0) = x0. Since any birational map with less than 3 fundamental points (including inﬁnitely near) is regular, we see that in the second case Φx0 (σ) is a projective automorphism. In the ﬁrst case, the image of the line q1, q2 is equal to the point px0 (σ(x0)). Thus Φx0 (σ) is not projective. Since it has at most 3 fundamental points, it must be a quadratic transformation.

Take two points x, y which do not lie on the same fibre of each projection 1 1 0 0 0 p1 : F0 → P , p2 : F0 → P . Let x = F1 ∩F2, y = F1 ∩F2, where F1,F1 are two 0 fibres of p1 and F2,F2 are two fibres of p2. Then tx,y is a birational automorphism of F0.

Proposition 5.4.3. Φx0 (tx,y) is a product of quadratic transformations. If x0 ∈

{x, y}, then Φx0 (tx,y) is a quadratic transformation. Otherwise, Φx0 (tx,y) is the product of two quadratic transformation.

Proof. Assume first that y is not infinitely near to x. Suppose x0 coincides with one of the points x, y, say x0 = x. It follows from Remark 5.4.1 that Φx0 (T ) is defined outside q1, q2, px0 (y). On the other hand, the image of the line q1, px0 (y) is 1 a point. Here we assume that the projection F0 → P is chosen in such a way that −1 its fibres are the proper transforms of lines through q1 under px0 . Thus Φx0 (T ) is not regular with at most three F -points, hence is a quadratic transformation. If x0 6= x, y, we compose tx,y with an automorphism σ of Q such that σ(x0) = x. Then

−1 Φx0 (tx,y ◦ σ) = Φx0 (tx0,σ (y)) = Φx0 (tx,y) ◦ Φx0 (σ).

By the previous lemma, Φx0 (σ) is a quadratic transformation. By the previous −1 argument, Φx0 (tx0,σ (y)) is a quadratic transformation. Also the inverse of a quadratic transformation is a quadratic transformation. Thus Φx0 (tx,y) is a product of two quadratic transformations. Now assume that y x. Take any point z 6= x. Then one can easily checks that tx,y = tz,y ◦ tx,z. Here we view y as an ordinary point on tx,z(F0).

Theorem 5.4.4. Any Cremona transformation of the plane is a composition of projective transformations and the standard quadratic transformation T0.

Proof. Combining the previous propositions, we obtain that any Cremona transformation of the plane is a composition of projective and quadratic transformations. A quadratic transformation with reduced base locus is a composition of the standard Cremona transformation Tst and a projective transformation. 120 LECTURE 5. FACTORIZATION PROBLEM

It is enough to show that a quadratic transformations T with two or one base points is a composition of Tst and projective transformations. Suppose T has fundamental points at p1, p2 and an inﬁnitely near point p3 1 p1. Choose a point q different from p1, p2, p3 and not lying on the line p1, p2. Let f be a quadratic transformation with base points at p1, p2, q. It is easy to check that f ◦ T is a quadratic transformation with three base points (p1, p2,T (q)). Composing it with projective automorphisms we get the standard quadratic transformation Tst. Finally, let us consider a quadratic transformation T with base points p3 p2 p1. Take a point q which is not on the line passing through p1 with tangent direction p2. Consider a quadratic transformation f with base points p1, p2, q. It is easy to see that f ◦ T is a quadratic transformation with F -points p1, p2, τ3(q). By the previous case we can write this transformation as a composition of Tst and projective transformation.

5.5 Hilda Hudson’s Theorem

Noether’s Theorem has no analogs in higher dimension. In fact, the following negative result was proven by Hilda Hudson 1927. Theorem 5.5.1. Any set of generators of the Cremaon group Cr(3) must contain uncountable number of transformations of degree d > 1. By a different method, and for arbitrary n ≥ 3, this result was proven recently by Ivan Pan. We will reproduce it here. First we generalize the construction of a dilated Cremona transformation. Obviously, the Cremona transformation deﬁned by (F0, QG1, . . . , QGn) contains the hypersurface V (Q) in its P-locus. Conversely, for any hypersurface V (Q0) of degree k with a point q of multiplicity ≥ k − 1 one can construct, using the previous lemma, a Cremona transformation of degree d > k that contains the hypersurface in its P-locus. For example, we can take Gi = ti, i = 1, . . . , n, d−k−1 0 Q = H Q , where V (H) is a general hyperplane containing q, and F0 is a 0 hypersurface of degree d with d − 1-multiple point at q with (F0,HQ ) = 1. Now we are ready to prove Hudson-Pan’s Theorem. Consider a family of un- countably many degree d > 1 hypersurfaces Xi, i ∈ I, no two birationally isomorphic, which all vanish at some point q with multiplicity d (for example, cones over plane pairwise non-isomorphic cubic curves3). For each such i ∈ I we construct a Cremona transformation Ti which contains Xi in its P-locus. Suppose we have a set of generators of Cr(n) which contains only countably many non-projective transformations. Each generator contains only ﬁnitely many hypersurfaces in its

3Here we use that n > 2 5.5. HILDA HUDSON’S THEOREM 121

P-locus. If we have a product of generators g = gi1 ··· gis , then each irreducible hypersurface in the P -locus of g is contained in the P -locus of some gik . Thus the set of hypersurfaces contained in the P-locus of some Cremona transformation is countable. This contradictions proves the theorem.