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

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 Content: Introduction -- Preliminaries -- Singularities of pairs -- Noether - Fano inequalities -- Auxiliary results -- Icosahedral group -- Basic properties -- Surfaces with icosahedral symmetry -- Quintic del Pezzo threefold -- Anticanonical linear system -- Combinatorics of lines and conics -- Special invariant curves -- Two Sarkisov links -- Invariant subvarieties -- Invariant cubic hypersurface -- Curves of low degree -- Orbits of small length -- Further properties of the invariant cubic -- Summary of orbits, curves, and surfaces -- Singularities of linear systems -- Base loci of invariant linear systems -- Proof of the main result -- Halphen pencils and elliptic fibrations. 1. Preliminaries -- 2. Icosahedral Group -- 3. Quintic Del Pezzo Threefold -- 4. Invariant Subvarieties -- 5. Singularities Of Linear Systems. Ivan Cheltsov, Constantin Shramov. A Chapman & Hall Book. Includes Bibliographical References And Index.