On a conjecture of Beltrametti and Sommese

TL;DR

This paper proves a weak version of a conjecture relating to the existence of global sections of certain divisors on projective manifolds, with specific results in three dimensions confirming a stronger non-vanishing conjecture and applications to Seshadri constants.

Contribution

It establishes a weak version of Beltrametti and Sommese's conjecture in all dimensions and confirms a stronger non-vanishing conjecture in three dimensions, with applications.

Findings

Proved a weak version of the conjecture in arbitrary dimension.

Confirmed the non-vanishing conjecture in dimension three.

Applied results to Seshadri constants.

Abstract

Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that $K_X+(n-1)A$ is nef, then $K_X+(n-1)A$ has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.