Birationally rigid Fano varieties. (English) Zbl 1145.14032
Russ. Math. Surv. 60, No. 5, 875-965 (2005); translation from Usp. Mat. Nauk 60, No. 5, 71-160 (2005).
This is a survey of the theory of birationally rigid Fano varieties. In the Introduction the notions of nonrationality, maximal singularity, and birational rigidity are discussed. Part 1 contains the following preliminaries: minimal model programme; log minimal model programme; movable log pairs; Noether–Fano–Iskovskikh inequality; cubic surfaces; Sarkisov programme; log adjunction and connectedness. In Part 2 the discussion of threefolds includes: quartic threefolds; sextic double solid; double cover of a quadric; intersection of a quadric and a cubic; weighted hypersurfaces. In Part 3 the following topic on higher-dimensional varieties are discussed: hypersurfaces; complete intersections; double spaces; triple spaces; cyclic covers.
Reviewer: Vladimir L. Popov (Moskva)
MSC:
14J45 | Fano varieties |
14E05 | Rational and birational maps |
14E30 | Minimal model program (Mori theory, extremal rays) |
14G22 | Rigid analytic geometry |
14J30 | \(3\)-folds |
14E07 | Birational automorphisms, Cremona group and generalizations |
14M20 | Rational and unirational varieties |