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Fano varieties. (English) Zbl 0912.14013
Parshin, A. N. (ed.) et al., Algebraic geometry V: Fano varieties. Transl. from the Russian by Yu. G. Prokhorov and S. Tregub. Berlin: Springer. Encycl. Math. Sci. 47, 1-245 (1999).
This excellent monograph aims to gather together the main known (both classical and modern) results and methods concerning the classification of Fano varieties. Fano varieties are the generalization in higher dimensions of Del Pezzo surfaces, and play a very important role in the classification of projective manifolds in dimension \(\geq 3\). More precisely, Mori’s theory of minimal models usually produces fibrations \(f:X\to Y\), with \(\dim(X)> \dim(Y)\), whose general fibre is a Fano variety. In the last decade Mori’s program has been completely carried out by Mori himself in dimension \(3\). There are two main aspects of the theory of Fano varieties: the biregular classification and questions related to rationality and unirationality (the latter aspect is usually more difficult). Moreover, motivated by the above-mentioned minimal model program, a larger class of Fano varieties has to be considered, namely the so-called \(\mathbb Q\)-Fano varieties. These are Fano varieties with some admissible singularities, and their study raises new difficult questions. In the book under review all these aspects are considered. Even if the monograph is not completely selfcontained, the authors present the main ideas and methods of the theory. They include proofs of the most important results. The book contains an extensive bibliography.
Here is briefly the contents of the book. The first chapter is a brief exposition of the Mori’s theory of minimal models. Chapter 2 gives the basic definitions, examples and the simple properties of Fano varieties. Chapter 3 is devoted to some results of Fujita and Iskovkikh concerning the biregular classification of \(n\)-dimensional Fano varieties of index \(n-1\) (sometimes also called Del Pezzo varieties). The fourth chapter deals with the biregular classification of Fano varieties with Picard number \(\rho=1\), and the fifth chapter with the approach due to Gushel and Mukai of studying Fano varieties with \(\rho=1\) using vector bundles. In chapter 6 the authors treat problems related to uniruledness, rational connectivity and boundedness of the degree of Fano varieties. Chapter 7 is devoted to the classification (due to Mori and Mukai) of Fano varieties with \(\rho\geq 2\). The problems of rationality and uniruledness are discussed in chapters 8-10. In chapter 11 some generalizations of Fano varieties are presented. The last chapter is an appendix containing the classification tables of Fano varieties.
All in all this book (whose authors have major contributions in the classification of Fano varieties) is very useful to anyone interested in the classification of projective manifolds.
For the entire collection see [Zbl 0903.00014].

14J45 Fano varieties
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)