On Fano varieties with large pseudo-index

TL;DR

This paper proves that Fano varieties with isolated quotient singularities and sufficiently large intersection with the anticanonical divisor have Picard number one, extending understanding of their geometric structure.

Contribution

It establishes a new criterion linking the intersection number with the anticanonical divisor to the Picard number for Fano varieties with singularities.

Findings

If $C ullet (-K_X) > rac{n}{2}+1$ or $> rac{2n}{3}$ for all curves, then the Picard number $ ho_X=1$.

The result applies to Fano varieties with isolated quotient singularities.

Provides a threshold condition for the Picard number based on intersection numbers.

Abstract

Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$ for every curve $C \subset X$, then $\rho_X=1$.