Fano 4-folds with rational fibrations

Casagrande, Cinzia

doi:10.2140/ant.2020.14.809

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Fano 4-folds with rational fibrations

Cinzia Casagrande

Vol. 14 (2020), No. 3, 787–813

DOI: 10.2140/ant.2020.14.809

Abstract

We study (smooth, complex) Fano 4-folds $X$ having a rational contraction of fiber type, that is, a rational map $X --\to Y$ 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 $ℙ^{1}$ or $ℙ^{2}$ , then the Picard number $ρ_{X}$ 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 $X --\to Z$ , regular and proper on an open subset of $X$ , with $dim (Z) = 3$ , then either $X$ is a product of surfaces, or $ρ_{X}$ is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

Keywords

Fano 4-folds, Mori dream spaces, birational geometry, MMP

Mathematical Subject Classification 2010

Primary: 14J45

Secondary: 14E30, 14J35

Milestones

Received: 27 February 2019

Revised: 17 September 2019

Accepted: 8 November 2019

Published: 1 June 2020

Authors

Cinzia Casagrande
Dipartimento di Matematica Università di Torino Italy

Vol. 14, No. 3, 2020