Basepoint Freeness for Nef and Big Line Bundles in Positive Characteristic

A general eventual freeness result is given for nef and big line bundles in positive characteristic.One application is to the geometry of the universal stable curve over M g , specifically, the eventual freeness of the relative dualizing sheaf, in positive characteristic.An example is given which shows this fails 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.