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 = jOY (1) := jH (Y; OY (1))j. 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 2 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 jV j ⊂ jLj 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) ! jV _j, and by restriction, the map dom(f) ! jV 0_j which factors through Y,! jV 0_j. 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 jHY j. 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 jV 0j ⊂ jL0j is a linear system on Y , then we define its inverse transform f −1(jV 0j) of jV 0j following the procedure from above defining the linear sys- −1 0 tem HX . The members of f (jV j) are the inverse transforms of members of jV 0j. When f is a morphism, the inverse transform is equal to the full transform f ∗(jV j) ⊂ jf ∗L0j. 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(jV j) and is called the base ideal of the linear system jV j. The closed subscheme Bs(HX ) of X defined by this ideal is called the base locus scheme of jV j. We have Bs(jV j) = \D2HX 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 jV j (equivalently, in the divisors D0;:::;Dr) is called the fixed component of jV j. 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 2 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 jV j ⊂ jL(−F )j is irreducible and defines a rational map f : X 99K jV j equal to the composition of f with the transpose isomorphism jV _j ! jV 0_j. The linear system is called basepoint-free, or simply free if its base scheme is ∼ empty, i.e. b(jV j) = 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 n Bs(HX ) ! jV j defined by the linear system jV j is the projection −1 ∼ ∼ _ _ Proj Sym(V ⊗ (LjU) ) = Proj Sym(V ⊗ OU ) = U × jV j ! jV j: If jV j is an irreducible linear system, dom(f) = X n Bs(HX )red = X n Supp(Bs(HX )): Let f : X 99K Y be a rational map defined by the inverse transform HX = jV j 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) jV j ⊂ jL ⊗ b(jV j)j := jH0(X; b(jV j) ⊗ L)j. (ii) Let φ : X0 ! X be a regular map, and V 0 = φ∗(V ) ⊂ H0(X0; φ∗L). Then −1 −1 φ (b(jV j)) = b(f (jV j)). 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(jV j) is an invertible ideal (i.e. isomorphic to OX (−F ) for some effec- tive divisor F ), then dom(f) = X and f is defined by the linear system jL(−F )j. (iv) If dom(f) = X, then b(jV j) 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 vari- eties 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 1 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 excep- tional 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.