A remark on generalized abundance for surfaces

Claudio Fontanari

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arXiv · 2307.15686

A remark on generalized abundance for surfaces

Claudio Fontanari

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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 $n \ge 3$.

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Claudio Fontanari. 2023-07-28. A remark on generalized abundance for surfaces. https://arxiv.org/abs/2307.15686

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