The gamma construction and asymptotic invariants of line bundles over arbitrary fields

TL;DR

This paper generalizes the theory of asymptotic invariants of line bundles from complex projective varieties to those over arbitrary fields, employing a scheme-theoretic gamma construction to handle imperfect fields.

Contribution

It introduces a scheme-theoretic gamma construction for arbitrary fields and extends key ampleness and base locus criteria to a broader class of varieties.

Findings

Extended ampleness criterion to arbitrary fields.

Generalized Nakayama's base locus description.

Reduced to F-finite fields using gamma construction.

Abstract

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, K\"uronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama's description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields.