The volume function is upper semicontinuous on families of divisors
Algebraic Geometry
2025-08-08 v2
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.
Cite
@article{arxiv.2504.16676,
title = {The volume function is upper semicontinuous on families of divisors},
author = {Junpeng Jiao},
journal= {arXiv preprint arXiv:2504.16676},
year = {2025}
}
Comments
Sections 4 and 5 have been removed, as the questions they addressed have now been fully resolved via a different method in my new preprint