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 $\mathscr{ M}_{ g, n}.$
Categories: math.AG math.NT