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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 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.
Algebraic varieties are shapes defined by polynomial equations.
Smooth Fano threefolds are a fundamental subclass that can be
thought of as higher-dimensional generalizations of ordinary
spheres. They belong to 105 irreducible deformation families. This
book determines whether the general element of each family admits a
Kähler–Einstein metric (and for many families, for all
elements), addressing a question going back to Calabi 70 years ago.
The book's solution exploits the relation between these metrics and
the algebraic notion of K-stability. Moreover, the book presents
many different techniques to prove the existence of a
Kähler–Einstein metric, containing many additional relevant
results such as the classification of all Kähler–Einstein smooth
Fano threefolds with infinite automorphism groups and computations
of delta-invariants of all smooth del Pezzo surfaces. This book
will be essential reading for researchers and graduate students
working on algebraic geometry and complex geometry.
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