Basepoint freeness for nef and big line bundles in positive characteristic.(English)Zbl 0954.14004

Summary: 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, 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.

MSC:

 14C20 Divisors, linear systems, invertible sheaves 14H30 Coverings of curves, fundamental group 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14G15 Finite ground fields in algebraic geometry
Full Text: