The Augmented Base Locus in Mixed Characteristic

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 $0$.