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Quasi-elementary contractions of Fano manifolds

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  01 November 2008

Cinzia Casagrande
Cinzia Casagrande*
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy (email: [email protected])
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Abstract

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Let X be a smooth complex Fano variety. We study ‘quasi-elementary’ contractions of fiber type of X, which are a natural generalization of elementary contractions of fiber type. If f:X→Y is such a contraction, then the Picard numbers satisfy ρX≤ρY+ρF, where F is a general fiber of f. We show that, if dim Y ≤3 and ρY≥4, then Y is smooth and Fano; if moreover ρY≥6, then X is a product. This yields sharp bounds on ρX when dim X=4 and X has a quasi-elementary contraction of fiber type, and other applications in higher dimensions.

Keywords

Fano varietiesMori theoryPicard number

MSC classification

Secondary: 14J45: Fano varieties 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 6 , November 2008 , pp. 1429 - 1460
Copyright
Copyright © Foundation Compositio Mathematica 2007