  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.
