On the Picard number of singular Fano varieties

TL;DR

This paper investigates the Picard number of certain singular Fano varieties, establishing bounds and structural properties related to their singularities and morphisms, especially in three dimensions.

Contribution

It provides new bounds on the Picard number difference involving prime divisors and characterizes the structure of Fano varieties with high Picard number.

Findings

Bound rho_X - rho_D < 9 for canonical singularities.

Existence of finite morphisms to products involving surfaces when rho_X - rho_D > 3.

In three dimensions, Picard number rho_X is at most 10.

Abstract

Let X be a Q-factorial Gorenstein Fano variety. Suppose that the singularities of X are canonical and that the locus where they are non-terminal has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D < 9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there exists a finite morphism from X to S x Y, where S is a surface with rho_S at most 9. As an application we prove that, if X has dimension 3, then rho_X is at most 10.