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Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. (English) Zbl 0926.14003
Cambridge Tracts in Mathematics. 134. Cambridge: Cambridge University Press. viii, 254 p. (1998).
The classification of algebraic surfaces, developed by the Italian algebraic geometers at the end of the last century and beginning of this century, was the major contribution of the classical Italian school of algebraic geometry. A part of this classification aims to find in every birational class of surfaces a “simplest” surface (called minimal model, which in most cases turns out to be unique). One of the most important discoveries in the last twenty years in algebraic geometry was that the theory of minimal models in dimension \(2\) can be generalized to higher dimensions. This is the so-called minimal, or Mori program. So far this program is completely satisfactory in dimension \(3\), although many important theorems are valid in arbitrary dimension. This theory turned out to have applications to various questions in algebraic geometry and in other fields. The work of Mori and Reid of the early 1980s suggested that a good theory of minimal models in dimension \(\geq 3\) should allow projective varieties with certain mild singularities. This was one of the major psychological difficulties the algebraic geometers had to overcome in order to build up a good theory of minimal models in general. The aim of the excellent book under review is to provide an introduction to the methods and ideas of the minimal model program in dimension \(\geq 3\). Due to the complexity of many techniques and proofs, the authors made a considerable effort to make the book as selfcontained as possible. This was done especially by simplifications of many proofs, and in case this was not quite possible, they tried to make a reasonable compromise between giving a complicated proof or explainaing it in a special (but relevant) case.
The book has seven chapters. Chapter 1 is an introduction to the minimal model program, and the results proved here provide the conceptual foundation for the whole book. Chapter 2 discusses the role of certain classes of singularities and some generalizations of the Kodaira vanishing theorem. In chapter 3 the cone theorem (valid in all dimensions) is proved. This is the first important ingredient of the theory. The last three chapters deal with the \(3\)-dimensional flips and flops. These are essentially new birational transformations in order to reach the minimal model. A special attention is devoted to some special classes of surface singularities or to the singularities occurring in the minimal model program in dimension \(3\). Chapter 6 is devoted to flops (which are easier to be understood than flips), and the last chapter to \(3\)-dimensional flips. The theory of \(3\)-dimensional flips is technically the most complicated part, and precisely here the authors manage to simplify considerably many proofs.
All in all this book will be of great help to those who want to introduce themselves to the beautiful and important theory of minimal models.

14E30 Minimal model program (Mori theory, extremal rays)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14E05 Rational and birational maps
14J30 \(3\)-folds
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14J40 \(n\)-folds (\(n>4\))