Numerical invariants of Fano 4-folds

TL;DR

This paper introduces an invariant for Fano 4-folds based on prime divisors and their cycle classes, establishing bounds on the Picard number related to this invariant.

Contribution

It defines a new integral invariant c_X for Fano 4-folds and proves bounds on the Picard number when c_X equals 2, extending previous classifications.

Findings

If c_X=2, then the Picard number rho_X is at most 12.

Previously known bounds for c_X>2 are confirmed, with c_X=3 implying rho_X at most 6.

The invariant c_X relates to the structure and classification of Fano 4-folds.

Abstract

Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at most 6. In this paper we show that if c_X=2, then rho_X is at most 12.