Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T03:29:55.333Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-elementary contractions of Fano manifolds

Part of: Birational geometry Surfaces and higher-dimensional varieties

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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