Cremona Groups and the Icosahedron Hardback
by Ivan (University of Edinburgh, Scotland) Cheltsov, Constantin (Steklov Institute of Mathematics, Moscow, Russia) Shramov
Part of the Chapman & Hall/CRC Monographs and Research Notes in Mathematics series
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.
- Format: Hardback
- Pages: 527 pages, No Color; 10 Tables, black and white; 36 Illustrations, black and white
- Publisher: Apple Academic Press Inc.
- Publication Date: 10/08/2015
- Category: Algebraic geometry
- ISBN: 9781482251593