The volume function is upper semicontinuous on families of divisors

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.

Publication Details

Journal
ArXiv.org
Published
2025-04-23
Primary Topic
Advanced Topology and Set Theory
Type
article
Field-Weighted Citation Impact
0.00

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article

The volume function is upper semicontinuous on families of divisors

ArXiv.org
Advanced Topology and Set Theory
article

The volume function is upper semicontinuous on families of divisors

article en

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.

ArXiv.org
Beijing Municipal Science and Technology Commission
Openalex Percentile: Top 42%
Advanced Topology and Set Theory
Open Access PDFOpenAlex
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The volume function is upper semicontinuous on families of divisors · ArXiv.org (2025) | TGRS Research Map | TGRS