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cremonalect - Lectures on Cremona transformations Ann...

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Lectures on Cremona transformations, Ann Arbor-Rome, 2010/2011 Igor Dolgachev 11th April 2011
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ii
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Contents 1 Basic properties 1 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 Jonqui`eres involutions . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Planar Cremona involutions . . . . . . . . . . . . . . . . . . . . . 78 4.3 De Jonqui`eres subgroups . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Linear systems of isologues . . . . . . . . . . . . . . . . . . . . . 81 iii
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iv CONTENTS 4.5 Arguesian involutions . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Geiser and Bertini involutions in P 3 . . . . . . . . . . . . . . . . 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
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Lecture 1 Basic properties 1.1 Generalities about rational maps and linear systems Recall that a rational map f : X 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 Rat K 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 f d to denote the restriction of f to dom ( f ) , or to any open subset of dom ( f ) . A dominant map f d : 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 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 Y is defined by a linear system. Namely, we embed Y in a projective space P r and consider the complete linear system H Y = |O Y (1) := | H 0 ( Y, O Y (1)) | . Its divisors are hyperplane sections of Y . The invertible sheaf f * d O Y (1) on dom ( f ) can be extended to a unique invertible sheaf L on all of X . Also we can extend the sections f * d ( s ) , s V , to sections of L on all of X . The obtained homomorphism f * : V = H 0 ( Y, O Y (1)) H 0 ( X, L ) is injective and its image is a linear subspace V H 0 ( X, L ) . The associated projective space
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