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

TL;DR

This paper investigates the properties of Moriwaki divisors on the moduli space of curves, linking augmented base loci to boundary conditions, and explores implications for the minimal model program and log canonical models.

Contribution

It establishes a characterization of Moriwaki divisors via augmented base loci and derives consequences for the geometry and minimal model program of ar{M}_g.

Findings

Moriwaki inequalities are equivalent to augmented base loci containment in the boundary.

Results impact the understanding of Zariski decompositions on ar{M}_g.

Findings influence the minimal model program and log canonical models of ar{M}_g.

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(\alpha).