## Abstract

Let M_{g, n} be the moduli space of n-pointed stable genus g curves, and let 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 MV generated by a stable S-module V, allowing to effectively compute S_{n}-equivariant Euler characteristics of M_{g, n} in terms of S_{n'}-equivariant Euler characteristics of M_{g', n'} with 0 ≤ g' ≤ g and max[0; 3-2g'] ≤ n' ≤ 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 MV effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the S_{n}-equivariant Euler characteristics of M_{g, n}, for g = 0; 1 and 2, in terms of the corresponding generating series associated with M_{g, n}.

Original language | English (US) |
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Pages (from-to) | 523-543 |

Number of pages | 21 |

Journal | Algebraic Geometry |

Volume | 7 |

Issue number | 5 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© Foundation Compositio Mathematica 2020.

## Keywords

- Asymptotic expansion
- Equivariant euler characteristics
- Free modular operad
- Moduli space
- Plethystic exponential

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