The augmented base locus in positive characteristic

TL;DR

This paper proves that in positive characteristic, the augmented base locus of a nef line bundle on a projective scheme equals the union of irreducible closed subsets where the bundle isn't big, extending known results from characteristic zero.

Contribution

The paper establishes a characteristic p version of Nakamaye's theorem relating augmented base loci and bigness of line bundles.

Findings

Augmented base locus equals union of non-big restrictions

Extends Nakamaye's theorem to positive characteristic

Provides a new geometric characterization in positive characteristic

Abstract

Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that the restriction of L to V is not big. For a smooth variety in characteristic zero, this was proved by Nakamaye using vanishing theorems.