cremonalect - Lectures on Cremona transformations, Ann...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lectures on Cremona transformations, Ann Arbor-Rome, 2010/2011 Igor Dolgachev 11th April 2011 ii 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 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 Hudsons 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 Rat....
View Full Document

Page1 / 125

cremonalect - Lectures on Cremona transformations, Ann...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online