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

Abstract

In this paper we study the ample cone of the moduli space M ¯ g , n \overline {M}_{g,n} of stable n n -pointed curves of genus g g . Our motivating conjecture is that a divisor on M ¯ g , n \overline {M}_{g,n} is ample iff it has positive intersection with all 1 1 -dimensional strata (the components of the locus of curves with at least 3 g + n − 2 3g+n-2 nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1 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 g=0 . More precisely, there is a natural finite map r : M ¯ 0 , 2 g + n → M ¯ g , n r: \overline {M}_{ 0, 2g+n} \rightarrow \overline {M}_{g,n} whose image is the locus R ¯ g , n \overline {R}_{g,n} of curves with all components rational. Any 1 1 -strata either lies in R ¯ g , n \overline {R}_{g,n} or is numerically equivalent to a family E E of elliptic tails, and we show that a divisor D D is nef iff D ⋅ E ≥ 0 D \cdot E \geq 0 and r ∗ ( D ) r^{*}(D) is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of M ¯ g , n \overline {M}_{g,n} for g ≥ 1 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.