A generalized non-vanishing theorem on surfaces

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.

7 pages. Final version, to appear in Pure and Applied Mathematics Quarterly (special volume in honor of Caucher Birkar)