Carolina Araujo

Ivan Cheltsov

Constantin Shramov

Ana-Maria Castravet

Kento Fujita

Anne-Sophie Kaloghiros

Jesus Martinez-Garcia

Hendrik Süß

Nivedita Viswanathan

This book determines whether the general element of each family of Fano threefolds is K-polystable, a major problem in mathematics.

The Calabi Problem for Fano Threefolds

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.

Cremona Groups and the Icosahedron

Constantin Shramov

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