  A necessary and sufficient condition is given for semi-ampleness of a
numerically effective (nef) and big line bundle in positive characteristic. One
application is to the geometry of the universal stable curve over M_g,
specifically, the semi-ampleness of the relative dualizing sheaf, in positive
characteristic. An example is given which shows this and the semi-ampleness
criterion fail in characteristic zero. A second application is to Mori's
program for minimal models of 3-folds in positive characteristic, namely, to
the existence of birational extremal contractions.
