Nef line bundles over finite fields

TL;DR

The paper constructs specific nef line bundles over finite fields that challenge existing cohomology vanishing expectations and introduces a new vanishing theorem for strictly nef line bundles.

Contribution

It provides explicit examples of nef line bundles with non-vanishing first cohomology over finite fields and proves a new vanishing theorem for strictly nef line bundles.

Findings

Constructed nef line bundles with non-vanishing H^1 over finite fields

Extended examples to threefolds improving previous work

Proved a new vanishing theorem for strictly nef line bundles

Abstract

We use Totaro's examples of non-semiample nef line bundles on smooth projective surfaces over finite fields to construct nef line bundles for which the first cohomology group cannot be killed by any generically finite covers. This is used to show a similar example of a nef and big line bundle on a smooth projective threefold over a finite field. This improves some examples of Bhatt and answers some of his questions. Finally, we prove a new vanishing theorem for the first cohomology group of strictly nef line bundles on projective varieties defined over finite fields.