Cremona Groups and the Icosahedron

ISBN-13: 9781482251593 / Angielski / Twarda / 2015 / 528 str.

Ivan Cheltsov; Constantin Shramov

Opis

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 "V"5, and gives a proof of its A5-birational rigidity.

The authors explicitly describe many interesting A5-invariant subvarieties of "V"5, including A5-orbits, low-degree curves, invariant anticanonical "K"3 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 "V"5 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 "V"5.

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 "V"5.

Spis treści

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