1st Edition

Cremona Groups and the Icosahedron

By Ivan Cheltsov, Constantin Shramov Copyright 2016
    528 Pages 36 B/W Illustrations
    by Chapman & Hall

    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.

    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

    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.