Counting lattice points in compactified moduli spaces of curves

TL;DR

This paper develops a method to count lattice points in the compactified moduli space of stable genus g curves, linking enumeration to tautological intersection numbers and orbifold Euler characteristics, with a recursive formula for computation.

Contribution

It extends previous lattice point counting methods to the compactified moduli space, establishing a recursive formula for calculating associated polynomials.

Findings

Polynomials encode tautological intersection numbers

Constant term equals orbifold Euler characteristic

Recursive formula enables effective computation

Abstract

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli space and constant term the orbifold Euler characteristic of the compactified moduli space. We also prove a recursive formula which can be used to effectively calculate these polynomials.