Numerical criteria for divisors on Mg to be ample

The moduli space Mg,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. These are believed to determine all ample divisors: F−Conjecture. A divisor on Mg,n is ample if and only if it positively intersects the F−curves. In this paper the F−conjecture on Mg,n is reduced to showing that certain divisors in M0,N for N ≤ g + n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary (cf. Theorem 3.1). As an application of the reduction, numerical criteria are given which if satisfied by a divisor D on Mg, show that D is ample (cf. Corollaries 5.1,5.2, 5.3, 5.4, and 5.5). Additionally, an algorithm is described to check that a given divisor is ample (cf. Theorem/Algorithm 4.5). Using a computer program called The Nef Wizard, written by Daniel Krashen, one can use the criteria and the algorithm to verify the conjecture for low genus. This is done on Mg for g ≤ 24, more than doubling the known cases of the conjecture and showing it is true for the first genus such that Mg is known to be of general type.