The volume function is upper semicontinuous on families of divisors

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.

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