Fano 4-folds with rational fibrations

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 ${\mathbb{P}}^1$ or ${\mathbb{P}}^2$, then the Picard number ${\rho }_X$ 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 $dim\left(Z\right)=3$, then either $X$ is a product of surfaces, or ${\rho }_X$ is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.