A remark on generalized abundance for surfaces

TL;DR

This paper proves a semi-ampleness result for certain nef divisors on surface pairs, providing a partial confirmation of the generalized abundance conjecture in dimension two, with counterexamples in higher dimensions.

Contribution

It establishes that for surface pairs, if the canonical divisor plus boundary and an additional nef divisor are not proportional, then their sum is semiample, advancing the understanding of the generalized abundance conjecture.

Findings

Proves semi-ampleness of $K_X + riangle + L$ under specified conditions for surfaces.

Shows the result does not extend to higher dimensions through counterexamples.

Provides insight into the structure of nef divisors on surfaces in the context of the abundance conjecture.

Abstract

Let $(X, \Delta)$ be a projective klt pair of dimension $2$ and let $L$ be a nef $\mathbb{Q}$-divisor on $X$ such that $K_X + \Delta + L$ is nef. As a complement to the Generalized Abundance Conjecture by Lazi\'c and Peternell, we prove that if $K_X + \Delta$ and $L$ are not proportional modulo numerical equivalence, then $K_X + \Delta + L$ is semiample. An example due to Lazi\'c shows that this is no longer true in any dimension $n \ge 3$.