ABSTRACT

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 a

chapter 1|20 pages

- Introduction

part 1|2 pages

Part I - Preliminaries

chapter 2|16 pages

- Singularities of pairs

chapter 3|14 pages

- Noether–Fano inequalities

chapter 4|12 pages

- Auxiliary results

part 2|2 pages

Part II - Icosahedral group

chapter 5|28 pages

- Basic properties

chapter 6|56 pages

- Surfaces with icosahedral symmetry

part 3|2 pages

Part III - Quintic del Pezzo threefold

chapter 7|36 pages

- Quintic del Pezzo threefold

chapter 8|18 pages

- Anticanonical linear system

chapter 9|14 pages

- Combinatorics of lines and conics

chapter 10|18 pages

- Special invariant curves

chapter 11|40 pages

- Two Sarkisov links

part 4|2 pages

Part IV - Invariant subvarieties

chapter 12|32 pages

- Invariant cubic hypersurface

chapter 13|34 pages

- Curves of low degree

chapter 14|28 pages

- Orbits of small length

chapter 15|30 pages

- Further properties of the invariant cubic

chapter 16|8 pages

- Summary of orbits, curves, and surfaces

part 5|2 pages

Part V - Singularities of linear systems

chapter 17|28 pages

- Base loci of invariant linear systems

chapter 18|22 pages

- Proof of the main result

chapter 19|20 pages

Halphen pencils and elliptic fibrations