Essentials of the method of maximal singularities

Abstract

A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The principal elements of the method are considered: N{\" o}ther-Fano inequality, maximal cycles, infinitely near maximal singularities, exclusion and untwisting. In a detailed way the crucial technical points are discussed. We also give a new version of the proof of Sarkisov theorem which is ideologically more close to the original arguments of V.A.Iskovskikh and Yu.I.Manin.

LATEX, 25 pages, to appear in "Proceedings of Symposium in Algebraic Geometry", ed. A.Corti and M.Reid. The volume is to come out in 1999

Cited by in corpus (13)

• Birationally rigid Fano hypersurfaces
• The rationality problem for conic bundles
• Log canonical thresholds of certain Fano hypersurfaces
• Birationally rigid iterated Fano double covers
• Birational rigidity of a three-dimensional double cone
• Birational properties of pencils of del Pezzo surfaces of degree 1 and 2
• Birationally rigid varieties with a pencil of Fano double covers. III
• Birational automorphisms of a three-dimensional double quadric with an elementary singularity
• Birational automorphisms of nodal quartic threefolds
• Birationally rigid singular double quadrics and double cubics
• Birational rigidity and Q-factoriality of a singular double quadric
• Double spaces with isolated singularities
• On a rigidity criterion for del Pezzo fibrations over ${\mathbb{P}}^1$