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Quasi-elementary contractions of Fano manifolds. (English) Zbl 1158.14037
A Fano manifold $$X$$ is a smooth projective variety with ample anticanonical class. These are, conjecturally, the building block of uniruled varieties [Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem. Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006)]. The paper under review aims to bound the rank of Picard group of Fano manifold, over the complex numbers, see also [L. Bonavero, C. Casagrande, O. Debarre, S. Druel, Comment. Math. Helv. 78, No. 3, 601–626 (2003; Zbl 1044.14019); M. Andreatta, E. Chierici, G. Occhetta, Cent. Eur. J. Math. 2, No. 2, 272–293, electronic only (2004; Zbl 1068.14049)]. To approach this problem one looks for fiber type contractions $$f:X\to Y$$ onto a lower dimensional, possibly Fano variety, to apply an inductive argument. To make it work the author studies “quasi-elementary” contractions. Namely morphisms $$f$$ where all numerical classes of curves contracted by $$f$$ are contained in the general fiber. Via a clever use of Mori’s theory of extremal rays and covering families of rational curves many interesting properties of these contractions are proved. This allows the author to bound the Picard number under hypothesis on the existence of quasi elementary contractions. In particular nice results for 4-folds and 5-folds are proved. The paper is complemented with many examples of these contractions.

##### MSC:
 14J45 Fano varieties 14E30 Minimal model program (Mori theory, extremal rays) 14J35 $$4$$-folds
##### Keywords:
Fano varieties; Picard rank; quasi-elementary contractions
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