By Parshin, Shafarevich

The purpose of this survey, written by means of V.A. Iskovskikh and Yu.G. Prokhorov, is to supply an exposition of the constitution conception of Fano types, i.e. algebraic vareties with an plentiful anticanonical divisor. Such kinds evidently look within the birational class of types of detrimental Kodaira size, and they're very just about rational ones. This EMS quantity covers diversified ways to the category of Fano forms similar to the classical Fano-Iskovskikh ''double projection'' process and its ameliorations, the vector bundles process because of S. Mukai, and the tactic of extremal rays. The authors speak about uniruledness and rational connectedness in addition to fresh development in rationality difficulties of Fano types. The appendix comprises tables of a few sessions of Fano forms. This e-book can be very necessary as a reference and study consultant for researchers and graduate scholars in algebraic geometry.

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3. 3 is far beyond the scope of this paper. Nevertheless, there are at least three main steps in it which we want to highlight as they provide sources of interesting (and difficult) open problems. 1 Dg-categories. X1 X2 / to compare the functors F and ˆE . This is done by passing to dg-enhancements and using a celebrated result of Toën. A; B/ ! A; B/ of degree 1 compatible with the composition. A/ its homotopy category. A; B/. A/ has a natural structure of triangulated category. A dg-functor F W A !

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