Arithmetically nef line bundles

TL;DR

This paper introduces the concept of arithmetically nef line bundles, characterizes them via stable base loci, and explores conditions under which nef line bundles are also arithmetically nef, especially when the base locus has low dimension.

Contribution

It defines arithmetically nef line bundles, proves their characterization through stable base loci, and establishes conditions for nef line bundles to be arithmetically nef.

Findings

Arithmetically nef line bundles are characterized by their restriction to stable base loci.

Nef line bundles with stable base locus of dimension ≤1 are arithmetically nef.

Not all nef line bundles are arithmetically nef, highlighting a nuanced difference.

Abstract

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be "thickened", allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line bundle $L$ may be nef, but not arithmetically nef. We show that $L$ is arithmetically nef if and only if its restriction to its stable base locus is arithmetically nef. Consequently, if $L$ is nef and its stable base locus has dimension $1$ or less, then $L$ is arithmetically nef.