Abstract
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 $\overline M_g$, 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.
