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.
A bound on the number of points of a curve in projective space over a finite field
Singular curves over a finite field and with many points
Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields
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