Relative semiampleness in mixed characteristic

TL;DR

This paper proves that nef line bundles on proper schemes over excellent bases are semiample if and only if they are semiample in both characteristic zero and positive characteristic, extending known results to mixed characteristic.

Contribution

It generalizes the semi-ampleness criterion to mixed characteristic and develops a new understanding of the Picard functor's properties in this setting.

Findings

Nef line bundles are semiample iff semiample in both characteristic zero and positive characteristic.

Generalization of the Picard functor's perfection as an h-stack in positive characteristic.

Provides a new approach to semi-ampleness in mixed characteristic contexts.

Abstract

We show that a nef line bundle on a proper scheme over an excellent base is semiample if and only if it is semiample after restricting to characteristic zero and to positive characteristic. In the process of the proof, we provide a generalisation to mixed characteristic of the fact that the perfection of the Picard functor in positive characteristic is a stack in groupoids for the h-topology.