Cremona Groups and the Icosahedron

ISBN: 9781482251593 出版年：2015 页码：521 Cheltsov, Ivan Shramov, Constantin CRC Press

Introduction Conjugacy in Cremona groups Three-dimensional projective space Other rational Fano threefolds Statement of the main result Outline of the book Preliminaries Singularities of pairs Canonical and log canonical singularities Log pairs with mobile boundaries Multiplier ideal sheaves Centers of log canonical singularities Corti's inequality Noether-Fano inequalities Birational rigidity Fano varieties and elliptic fibrations Applications to birational rigidity Halphen pencils Auxiliary results Zero-dimensional subschemes Atiyah flops One-dimensional linear systems Miscellanea Icosahedral Group Basic properties Action on points and curves Representation theory Invariant theory Curves of low genera SL2(C) and PSL2(C) Binary icosahedral group Symmetric group Dihedral group Surfaces with icosahedral symmetry Projective plane Quintic del Pezzo surface Clebsch cubic surface Two-dimensional quadric Hirzebruch surfaces Icosahedral subgroups of Cr2(C) K3 surfaces Quintic del Pezzo Threefold Quintic del Pezzo threefold Construction and basic properties PSL2(C)-invariant anticanonical surface Small orbits Lines Orbit of length five Five hyperplane sections Projection from a line Conics Anticanonical linear system Invariant anticanonical surfaces Singularities of invariant anticanonical surfaces Curves in invariant anticanonical surfaces Combinatorics of lines and conics Lines Conics Special invariant curves Irreducible curves Preliminary classification of low degree curves Two Sarkisov links Anticanonical divisors through the curve L6 Rational map to P4 A remarkable sextic curve Two Sarkisov links Action on the Picard group Invariant Subvarieties Invariant cubic hypersurface Linear system of cubics Curves in the invariant cubic Bring's curve in the invariant cubic Intersecting invariant quadrics and cubic A remarkable rational surface Curves of low degree Curves of degree 16 Six twisted cubics Irreducible curves of degree 18 A singular curve of degree 18 Bring's curve Classification Orbits of small length Orbits of length 20 Ten conics Orbits of length 30 Fifteen twisted cubics Further properties of the invariant cubic Intersections with low degree curves Singularities of the invariant cubic Projection to Clebsch cubic surface Picard group Summary of orbits, curves, and surfaces Orbits vs. curves Orbits vs. surfaces Curves vs. surfaces Curves vs. curves Singularities of Linear Systems Base loci of invariant linear systems Orbits of length 10 Linear system Q3 Isolation of orbits in S Isolation of arbitrary orbits Isolation of the curve L15 Proof of the main result Singularities of linear systems Restricting divisors to invariant quadrics Exclusion of points and curves different from L15 Exclusion of the curve L15 Alternative approach to exclusion of points Alternative approach to the exclusion of L15 Halphen pencils and elliptic fibrations Statement of results Exclusion of points Exclusion of curves Description of non-terminal pairs Completing the proof