Effective non-vanishing conjectures for projective threefolds

TL;DR

This paper proves new non-vanishing results for adjoint bundles on smooth projective threefolds, confirming conjectures in specific cases and providing simplified proofs for known results.

Contribution

It establishes non-vanishing of certain adjoint bundles under conditions on the canonical bundle and offers a concise proof of the Beltrametti-Sommese conjecture in dimension three.

Findings

If K_X or -K_X is pseudoeffective, then K_X+A has global sections.

A short proof of the Beltrametti-Sommese conjecture for threefolds is provided.

The results confirm non-vanishing conjectures in specific cases for projective threefolds.

Abstract

Let X be a smooth projective threefold, and let A be an ample line bundle such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective, the adjoint bundle $K_X+A$ has global sections. We also give a very short proof of the Beltrametti-Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint bundle $K_X+2A$ has global sections.