BASEPOINT FREENESS FOR NEF AND BIG LINE BUNDLES IN POSITIVE CHARACTERISTIC, WITH APPLICATIONS TO Mg,n AND TO 3-FOLD MMP

A necessary and sufficient condition is given for semi-ampleness of a 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. §0 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 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). Indeed a number of the main results and conjectures –the Basepoint Free Theorem, the Abundance Conjecture, quasiprojectivity of moduli spaces – are explicitly issues of semi-ampleness. I will give some details below. There is a necessary numerical condition for semi-ampleness. The restriction of an semi-ample line bundle to a curve must have non-negative 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 [Kollár96,VI.2.17], nefness is equivalent to the apparently stronger condition: L · Z ≥ 0 for every proper irreducible Z ⊂ X . (I During this research I was partially supported by NSF grant DMS-9531940