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

Keel, Seán

Ann. Math. (2) 149, No. 1, 253-286 (1999).

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

Keywords:

nef line bundle; base point freeness; cone theorem; semi-ampleness; big line bundle; positive characteristic; Mori’s program; extremal contractions