Moriwaki divisors and the augmented base loci of divisors on the moduli space of curves

Abstract

We study the cone of Moriwaki divisors on \bar{M}_g by means of augmented base loci. Using a result of Moriwaki, we prove that an R-divisor D satisfies the strict Moriwaki inequalities if and only if the augmented base locus of D is contained in the boundary of \bar{M}_g. Then we draw some interesting consequences on the Zariski decomposition of divisors on \bar{M}_g, on the minimal model program of \bar{M}_g and on the log canonical models \bar{M}_g(α).

14 pages. v2: minor revision. v3: Statements of Corollaries 3 and 4 made more precise. Lemmas 2.1 and 2.3 stated and proved in arbitrary characteristic. Added Lemma 3.2. Added subsection 3.1 to explain in detail what is missing in order to extend our results over a field of positive characteristic. Final version, to appear in Michigan Mathematical Journal