Fano 4-folds with $b_2>12$ are products of surfaces

TL;DR

This paper proves that smooth complex Fano 4-folds with a Picard number greater than 12 are necessarily products of del Pezzo surfaces, expanding understanding of their structure based on contraction analysis.

Contribution

It establishes a classification criterion for Fano 4-folds with high Picard number, showing they decompose into products of surfaces, which was previously unknown.

Findings

Fano 4-folds with rho > 12 are products of del Pezzo surfaces.

Divisorial contractions lead to smooth del Pezzo surfaces.

The structure of Fano 4-folds is constrained by their Picard number.

Abstract

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the image S of the exceptional divisor is a surface, together with the author's previous work on Fano 4-folds. In particular, given f: X->Y as above, under suitable assumptions we show that S is a smooth del Pezzo surface with -K_S given by the restriction of -K_Y.