A generalized non-vanishing theorem on surfaces

TL;DR

This paper proves a key equivalence for the anti-canonical bundle on surfaces, linking numerical effectiveness and pseudo-effectiveness, and establishes a numerical non-vanishing theorem for polarized surfaces, answering a question by Fontanari.

Contribution

It introduces a new numerical non-vanishing theorem for surfaces with pseudo-effective divisors and characterizes the anti-canonical bundle's properties on $Q$-factorial surfaces.

Findings

Anti-canonical bundle is NE iff pseudo-effective on $Q$-factorial surfaces

Established a numerical non-vanishing theorem for polarized surfaces

Answered a question posed by C. Fontanari

Abstract

We show that the anti-canonical bundle of any $\mathbb Q$-factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.