Essentials of the method of maximal singularities

TL;DR

This paper provides a detailed exposition of the method of maximal singularities, a key technique in birational geometry, including new proofs and technical insights for understanding automorphisms of Fano varieties.

Contribution

It offers a comprehensive and detailed presentation of the method of maximal singularities, including a new version of the Sarkisov theorem proof closer to original ideas.

Findings

Clarifies the key elements of the method of maximal singularities

Provides a new proof of Sarkisov theorem aligned with original arguments

Discusses technical aspects like N{" o}ther-Fano inequality and maximal cycles

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.