Equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$

TL;DR

This paper develops an asymptotic expansion method to compute the equivariant Euler characteristics of the moduli space of stable pointed curves, linking them to those of smooth curves, and making previous theoretical representations effective.

Contribution

It provides an effective computational approach for the equivariant Euler characteristics of ar{\umathscr{M}}_{g, n} using asymptotic expansion, answering a question by Getzler and Kapranov.

Findings

Derived formulas for genus 0, 1, and 2 cases.

Linked stable and smooth curve Euler characteristics.

Made the integral representation of the modular operad characteristic effective.

Abstract

Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ $\textrm{max}\{0, 3 - 2g' \} \le n' \le 2(g - g') + n.$ This answers a question posed by Getzler and Kapranov by making their integral representation of the characteristic of the modular operad $\mathbb{M}\mathcal{V}$ effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n},$ for $g = 0, 1$ and $2,$ in terms of the corresponding generating series associated with $\mathscr{M}_{g, n}.$