Towards the ample cone of $\overline {M}_{g,n}$

Abstract:

In this paper we study the ample cone of the moduli space $\overline {M}_{g,n}$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\overline {M}_{g,n}$ is ample iff it has positive intersection with all $1$-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the $1$-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \overline {M}_{ 0, 2g+n} \rightarrow \overline {M}_{g,n}$ whose image is the locus $\overline {R}_{g,n}$ of curves with all components rational. Any $1$-strata either lies in $\overline {R}_{g,n}$ or is numerically equivalent to a family $E$ of elliptic tails, and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^{*}(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\overline {M}_{g,n}$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary.