# Towards the classification of Fano 4-folds with $b_2\geq 7$

## TL;DR

This paper classifies Fano 4-folds with high Picard number, showing they are either products of surfaces or related to cubic 4-folds, and explores their birational geometry and contractions.

## Contribution

It improves classification results for Fano 4-folds with Picard number greater than 6, providing explicit descriptions and new geometric insights.

## Key findings

- For Picard number >9, Fano 4-folds are products of del Pezzo surfaces.
- In the range 7 ≤ ρ(X) ≤ 9, non-product cases relate to blow-ups of cubic 4-folds.
- Describes the structure of Fano 4-folds with small elementary contractions.

## Abstract

We study (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. We show that if rho(X)>9, then X is a product of del Pezzo surfaces, thus improving recent results by the author and by the author and S.A. Secci; the statement is now optimal. In the range rho(X)=7,8,9 we show that if X is not a product of surfaces, and has no small elementary contraction, then it is the blow-up of a cubic 4-fold along a special configuration of planes. When instead rho(X)>6 and X has a small elementary contraction, we study X depending on its fixed prime divisors, giving explicit results on the geometry of X in the framework of birational geometry. In particular for the boundary case rho(X)=9 we show that either X is a product of surfaces, or X belongs to two explicit families, or there is a sequence of flips X-->X' where X' is a smooth projective 4-fold with an elementary contraction onto a 3-fold. In the paper we also give several results on rational contractions of fiber type of Fano 4-folds, and more generally of Mori dream spaces; in particular we use some properties of del Pezzo surfaces over non-closed fields, applied to generic fibers.