  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.
