# Fano 4-folds with rational fibrations

## TL;DR

This paper investigates the structure of smooth Fano 4-folds with rational fibrations, establishing bounds on their Picard number and characterizing cases where the bounds are attained, advancing the understanding of their birational geometry.

## Contribution

It provides new bounds on the Picard number of Fano 4-folds with rational fibrations and characterizes when these bounds are achieved, especially in relation to product structures.

## Key findings

- If Y is not P^1 or P^2, then rho(X) ≤ 18, with equality only if X is a product of surfaces.
- If X has a dominant rational map to a 3-dimensional Z, then either X is a product of surfaces or rho(X) ≤ 12.
- The results contribute to classifying Fano 4-folds with large Picard number via birational geometry.

## 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 non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or 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-->Z, regular and proper on an open subset of X, with dim(Z)=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.