Volumes of line bundles as limits on generically nonreduced schemes

TL;DR

This paper investigates the conditions under which the volume of a line bundle, defined via a limsup, actually converges as a limit on certain nonreduced schemes, extending known results beyond generically reduced schemes.

Contribution

It proves that volumes are limits on specific classes of nonreduced schemes, including certain codimension one subschemes and schemes with nilradical squared zero.

Findings

Volumes are limits on codimension one subschemes with normal points.

Volumes are limits on schemes with nilradical squared zero.

Extends the class of schemes where volume limits are known to exist.

Abstract

The volume of a line bundle is defined in terms of a limsup. It is a fundamental question whether this limsup is a limit. It has been shown that this is always the case on generically reduced schemes. We show that volumes are limits in two classes of schemes that are not necessarily generically reduced: codimension one subschemes of projective varieties such that their components of maximal dimension contain normal points and projective schemes whose nilradical squared equals zero.