1st Edition

Cremona Groups and the Icosahedron




ISBN 9781482251593
Published August 21, 2015 by Chapman and Hall/CRC
527 Pages 36 B/W Illustrations

USD $115.00

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Book Description

Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity.

The authors explicitly describe many interesting A5-invariant subvarieties of V5, including A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular surface of general type that is a degree five cover of the diagonal Clebsch cubic surface. They also present two birational selfmaps of V5 that commute with A5-action and use them to determine the whole group of A5-birational automorphisms. As a result of this study, they produce three non-conjugate icosahedral subgroups in the Cremona group of rank 3, one of them arising from the threefold V5.

This book presents up-to-date tools for studying birational geometry of higher-dimensional varieties. In particular, it provides readers with a deep understanding of the biregular and birational geometry of V5.

Table of Contents

Introduction
Conjugacy in Cremona groups
Three-dimensional projective space
Other rational Fano threefolds
Statement of the main result
Outline of the book

Preliminaries
Singularities of pairs

Canonical and log canonical singularities
Log pairs with mobile boundaries
Multiplier ideal sheaves
Centers of log canonical singularities
Corti’s inequality

Noether–Fano inequalities
Birational rigidity
Fano varieties and elliptic fibrations
Applications to birational rigidity
Halphen pencils

Auxiliary results
Zero-dimensional subschemes
Atiyah flops
One-dimensional linear systems
Miscellanea

Icosahedral Group
Basic properties

Action on points and curves
Representation theory
Invariant theory
Curves of low genera
SL2(C) and PSL2(C)
Binary icosahedral group
Symmetric group
Dihedral group

Surfaces with icosahedral symmetry
Projective plane
Quintic del Pezzo surface
Clebsch cubic surface
Two-dimensional quadric
Hirzebruch surfaces
Icosahedral subgroups of Cr2(C)
K3 surfaces

Quintic del Pezzo Threefold
Quintic del Pezzo threefold
Construction and basic properties
PSL2(C)-invariant anticanonical surface
Small orbits
Lines
Orbit of length five
Five hyperplane sections
Projection from a line
Conics

Anticanonical linear system
Invariant anticanonical surfaces
Singularities of invariant anticanonical surfaces
Curves in invariant anticanonical surfaces

Combinatorics of lines and conics
Lines
Conics

Special invariant curves
Irreducible curves
Preliminary classification of low degree curves

Two Sarkisov links
Anticanonical divisors through the curve L6
Rational map to P4
A remarkable sextic curve
Two Sarkisov links
Action on the Picard group

Invariant Subvarieties
Invariant cubic hypersurface
Linear system of cubics
Curves in the invariant cubic
Bring’s curve in the invariant cubic
Intersecting invariant quadrics and cubic
A remarkable rational surface

Curves of low degree
Curves of degree 16
Six twisted cubics
Irreducible curves of degree 18
A singular curve of degree 18
Bring’s curve
Classification

Orbits of small length
Orbits of length 20
Ten conics
Orbits of length 30
Fifteen twisted cubics

Further properties of the invariant cubic
Intersections with low degree curves
Singularities of the invariant cubic
Projection to Clebsch cubic surface
Picard group

Summary of orbits, curves, and surfaces
Orbits vs. curves
Orbits vs. surfaces
Curves vs. surfaces
Curves vs. curves

Singularities of Linear Systems
Base loci of invariant linear systems
Orbits of length 10
Linear system Q3
Isolation of orbits in S
Isolation of arbitrary orbits
Isolation of the curve L15

Proof of the main result
Singularities of linear systems
Restricting divisors to invariant quadrics
Exclusion of points and curves different from L15
Exclusion of the curve L15
Alternative approach to exclusion of points
Alternative approach to the exclusion of L15

Halphen pencils and elliptic fibrations
Statement of results
Exclusion of points
Exclusion of curves
Description of non-terminal pairs
Completing the proof

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Author(s)

Biography

Ivan Cheltsov is a professor in the School of Mathematics at the University of Edinburgh. Dr. Cheltsov’s research focuses on birational geometry and its connections with algebra, geometry, and topology, including del Pezzo surfaces, Fano threefolds, and Cremona groups.

Constantin Shramov is a researcher at Steklov Mathematical Institute and Higher School of Economics in Moscow. Dr. Shramov’s research interests include birational geometry, Fano varieties, minimal model program, log-canonical thresholds, Kahler–Einstein metrics, Cremona groups, and birational rigidity.