Algebra & Number Theory

Fano 4-folds with rational fibrations

Cinzia Casagrande

Abstract

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 $ℙ1$ or $ℙ2$, 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

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

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

https://projecteuclid.org/euclid.ant/1593655272

Digital Object Identifier
doi:10.2140/ant.2020.14.787

Mathematical Reviews number (MathSciNet)
MR4113781

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

Citation

Casagrande, Cinzia. Fano 4-folds with rational fibrations. Algebra Number Theory 14 (2020), no. 3, 787--813. doi:10.2140/ant.2020.14.787. https://projecteuclid.org/euclid.ant/1593655272

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