2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148414I give a necessary and sufficient condition for a nef and big line bundle in positive characteristic to be semi-ample, and then give two applications: I show that the relative dualizing sheaf of the universal curve is semi-ample, in positive characteristic, and give a simple example which shows that this, and the semi-ampleness criterion, fail in characteristic zero. My second application is to Mori's program for minimal models of 3-folds. I prove a version of the Basepoint Free Theorem (for big line bundles on 3-folds of positive characteristic) and a simple Cone Theorem, for 3-folds of positive Kodaira dimension.I have corrected several errors from the original version, and simplified the proof of the contraction theorem. to appear in Annals of Math. arXiv admin note: see math.AG/9901149 for published version accidentally submitted anew in Annals migrationAlgebraic Geometrytext