Algebra & Number Theory

Fano 4-folds with rational fibrations

Cinzia Casagrande

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We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X −−→Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not 1 or 2, then the Picard number ρX of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X −−→Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or ρX is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

Article information

Algebra Number Theory, Volume 14, Number 3 (2020), 787-813.

Received: 27 February 2019
Revised: 17 September 2019
Accepted: 8 November 2019
First available in Project Euclid: 2 July 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds

Fano 4-folds Mori dream spaces birational geometry MMP


Casagrande, Cinzia. Fano 4-folds with rational fibrations. Algebra Number Theory 14 (2020), no. 3, 787--813. doi:10.2140/ant.2020.14.787.

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