Numerical criteria for divisors on $\M_{g}$ to be ample

Abstract

The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional components of this stratification are smooth rational curves (whose numerical equivalence classes are) called F-curves. The F-conjecture asserts that a divisor on $\M_{g,n}$ is ample if and only if it positively intersects the $F-$curves. In this paper the F-conjecture on $\M_{g,n}$ is reduced to showing that certain divisors in $\M_{0,N}$ for $N \leq g+n$ are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. As an application of the reduction, numerical criteria are given which if satisfied by a divisor $D$ on $\M_g$, show that $D$ is ample. Additionally, an algorithm is described to check that a given divisor is ample. Using a computer program called Nef Wizard, written by Daniel Krashen (http://www.yale.edu/users/dkrashen/nefwiz), one can use the criteria and the algorithm to verify the conjecture for low genus. This is done on $\M_g$ for $g \le 24$, more than doubling the known cases of the conjecture and showing it is true for the first genus such that $\M_g$ is known to be of general type.