The Augmented Base Locus in Mixed Characteristic

TL;DR

This paper extends the known equality of augmented and exceptional base loci for nef and big line bundles from projective schemes over fields to schemes over more general bases, including mixed characteristic cases.

Contribution

It generalizes the equality of augmented and exceptional base loci to schemes over arbitrary excellent Noetherian bases, assuming characteristic zero results.

Findings

Equality holds over mixed characteristic Dedekind domains.

Results apply when the line bundle is semiample in characteristic zero.

Extends classical results to more general base schemes.

Abstract

Let $L$ be a nef and big line bundle on a scheme $X$. It is well known that if $X$ is a projective over a field then the augmented base locus and the exceptional base locus agree. This result is extended to projective schemes over arbitrary excellent Noetherian bases, assuming the result holds in characteristic zero. In particular the result holds if the base is a mixed characteristic Dedekind domain or if $L$ is semiample in characteristic $0$.