On the birational geometry of Fano 4-folds

TL;DR

This paper investigates the birational geometry of Fano 4-folds, focusing on rational contractions and their properties, establishing bounds on Picard numbers related to these contractions.

Contribution

It provides a characterization of non-movable prime divisors and establishes bounds on Picard numbers for Fano 4-folds with certain types of rational contractions.

Findings

Picard number at most 11 with elementary fiber type contractions

Picard number at most 18 with quasi-elementary fiber type contractions

Characterization of non-movable prime divisors for Picard number ≥ 6

Abstract

We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, quasi-elementary. The main result is that the Picard number of X is at most 11 if X has an elementary rational contraction of fiber type, and 18 if X has a quasi-elementary rational contraction of fiber type.