Counting lattice points in the moduli space of curves

TL;DR

This paper introduces a method to count lattice points in the moduli space of genus g curves with n marked points, linking combinatorial enumeration to geometric invariants like Euler characteristics and intersection numbers.

Contribution

It defines a novel counting approach for lattice points in moduli spaces, connecting combinatorial counts with geometric and topological invariants.

Findings

Produces a polynomial encoding Euler characteristics and intersection numbers

Establishes a new link between lattice point counting and moduli space invariants

Provides a framework for enumerating geometric structures in moduli spaces

Abstract

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.