Cremona Groups and the Icosahedron

1. Auflage 2015
ISBN: 978-1-4822-5159-3
Verlag: Chapman and Hall/CRC

Buch, Englisch, 528 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 951 g

Reihe: 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.

Zielgruppe

Mathematicians and graduate students in algebra, geometry, and number theory.

Autoren/Hrsg.

Cheltsov, Ivan

Shramov, Constantin

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

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

Über Autor(innen)

Ivan Cheltsov is a professor in the School of Mathematics at the University of Edinburgh. Dr. Cheltsov’s research focuses on birational geometry and its connections with algebra, geometry, and topology, including del Pezzo surfaces, Fano threefolds, and Cremona groups.

Constantin Shramov is a researcher at Steklov Mathematical Institute and Higher School of Economics in Moscow. Dr. Shramov’s research interests include birational geometry, Fano varieties, minimal model program, log-canonical thresholds, Kahler–Einstein metrics, Cremona groups, and birational rigidity.