Fano 4-folds, flips, and blow-ups of points

TL;DR

This paper investigates the structure of smooth Fano 4-folds with high Picard number, establishing bounds and providing examples, using birational geometry and the study of fixed prime divisors.

Contribution

It proves an upper bound on the Picard number for certain Fano 4-folds and explores the role of fixed prime divisors in their classification.

Findings

Picard number of such Fano 4-folds is at most 12.

Examples of Fano 4-folds with Picard number up to 9 are constructed.

General results on fixed prime divisors in Fano 4-folds are obtained.

Abstract

In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest.