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Cremona Groups and the Icosahedron (Hardcover) (Ivan Cheltsov & Constantin Shramov)
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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 ofV5, including A5-orbits, low-degree curves, invariant anticanonicalK3 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 ofV5 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 ofV5.
Focusing on Cremona groups and their icosahedral subgroups, this book provides an ideal introduction to birational geometry for researchers and graduate students. The first results of the Cremona group of rank two go back to the time of Appollonius. But the map of the Cremona group is not often defined at some points, making it very hard to study. This book introduces the Cremona group of rank three, which is very large and impossible to handle directly. The book helps readers develop new methods for the study of finite subgroups of Cremona groups of high rank or Cremona groups of rank three.