Counting lattice points in compactified moduli spaces of curves

Abstract

We define and count lattice points in the moduli space ${\overline{\mathcal{ℳ}}}_{g,n}$ of stable genus $g$ curves with $n$ labeled points. This extends a construction of the second author for the uncompactified moduli space ${\mathcal{ℳ}}_{g,n}$. The enumeration produces polynomials whose top degree coefficients are tautological intersection numbers on ${\overline{\mathcal{ℳ}}}_{g,n}$ and whose constant term is the orbifold Euler characteristic of ${\overline{\mathcal{ℳ}}}_{g,n}$. We prove a recursive formula which can be used to effectively calculate these polynomials. One consequence of these results is a simple recursion relation for the orbifold Euler characteristic of ${\overline{\mathcal{ℳ}}}_{g,n}$.