Cremona Groups and the Icosahedron: 1st Edition (Hardback) book cover

Cremona Groups and the Icosahedron

1st Edition

By Ivan Cheltsov, Constantin Shramov

Chapman and Hall/CRC

527 pages | 36 B/W Illus.

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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


Conjugacy in Cremona groups

Three-dimensional projective space

Other rational Fano threefolds

Statement of the main result

Outline of the book


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


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


Orbit of length five

Five hyperplane sections

Projection from a line


Anticanonical linear system

Invariant anticanonical surfaces

Singularities of invariant anticanonical surfaces

Curves in invariant anticanonical surfaces

Combinatorics of lines and 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


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

About the Authors

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.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Geometry / General
MATHEMATICS / Number Theory