Basepoint freeness for nef and big line bundles in positive characteristic

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 Mg, 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. Introduction and statement of results A map from a variety to projective space is determined by a line bundle and a collection of global sections with no common zeros. As all maps between projective varieties arise in this way, one commonly wonders whether a given line bundle is generated by global sections, or equivalently, if the associated linear system is basepoint free. Once a line bundle L has a section, one expects the positive tensor powers L⊗n to have more sections. If some such power is globally generated, one says that L is semi-ample. Semi-ampleness is particularly important in Mori’s program for the classification of varieties (also known as the minimal model program, MMP). Indeed a number of the main results and conjectures — the Basepoint Free Theorem, the Abundance Conjecture, quasi-projectivity of moduli spaces — are explicitly issues of semi-ampleness. Some details will be given below. There is a necessary numerical condition for semi-ampleness. The restriction of a semi-ample line bundle to a curve must have nonnegative degree; thus, if the line bundle L on X is semi-ample, then L is nef ; i.e., L ·C ≥ 0 for every irreducible curve C ⊂ X. By a result of Kleiman (see [Kol96,VI.2.17]), nefness is equivalent to the apparently stronger condition: LdimZ · Z ≥ 0 for every proper irreducible Z ⊂ X. (We note this in relation to (0.1) below.) *Partially supported by NSF grant DMS-9531940.