Cremona Groups and the Icosahedron (Chapman & Hall/crc Monographs and Research Notes in Mathematics)

Cheltsov, Ivan/ Shramov, Constantin

Chapman & Hall/CRC（2015/08発売）

• 製本 Hardcover:ハードカバー版／ページ数 528 p.
• 言語 ENG
• 商品コード 9781482251593
• DDC分類 516.35

Full 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.

Contents

Preliminaries. Icosahedral Group. Quintic del Pezzo Threefold. Invariant Subvarieties. Singularities of Linear Systems.