Volumes of Line Bundles on Schemes

TL;DR

This paper investigates the existence of volumes of line bundles as limits on various schemes, proving their existence on certain subschemes of nonsingular projective varieties.

Contribution

It extends the known existence of line bundle volumes as limits from generically reduced schemes to specific subschemes of nonsingular projective varieties.

Findings

Volumes exist as limits on certain subschemes of nonsingular projective varieties.

The result broadens understanding of volume limits beyond generically reduced schemes.

Abstract

Volumes of line bundles are known to exist as limits on generically reduced projective schemes. However, it is not known if they always exist as limits on more general projective schemes. We show that they do always exist as a limit on a codimension one subscheme of a nonsingular projective variety.