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arXiv · 2504.16676

The volume function is upper semicontinuous on families of divisors

Junpeng Jiao
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Abstract

We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function of a divisor is upper semicontinuous in flat families with reduced and irreducible fibers.

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Junpeng Jiao. 2025-08-07. The volume function is upper semicontinuous on families of divisors. https://arxiv.org/abs/2504.16676

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