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.
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