[Paper Review] Euler Characteristics of Moduli Spaces of Curves

Gilberto Bini, John Harer|ArXiv.org|Jun 5, 2005

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper computes the orbifold and ordinary Euler characteristics of the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g}^{n}$ of the moduli space of $n$-pointed genus $g$ Riemann surfaces for all $g \geq 0$ and $n > 2 - 2g$. Using generating functions, integral representations, and Wick’s lemma, the authors derive explicit formulae (Theorems 3.2, 4.3, and 4.5), confirming and extending previous results for low genera and resolving discrepancies in earlier computations for $g=2$. The key contribution is a unified, algorithmic method for computing these topological invariants across all genera and puncture numbers.

ABSTRACT

Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.

Motivation & Objective

• To compute the orbifold and ordinary Euler characteristics of the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g}^{n}$ for all $g \geq 0$ and $n > 2 - 2g$.
• To extend previous results on the Euler characteristic of $\overline{\mathcal{M}}_{g}^{n}$, particularly for $g=2$, where discrepancies with earlier work were identified.
• To provide a systematic method using generating functions and integral representations that unifies computation across all genera and puncture numbers.
• To resolve an error in the generating function for $g=2$ reported in [2], correcting the coefficient of $D^5$ and aligning results with the authors' new formulae.

Proposed method

• The authors use stable graphs to stratify $\overline{\mathcal{M}}_{g}^{n}$, assigning each stratum $\Delta_G^o$ to a graph $G$ with vertex genus, edge connections, and marked points.
• They compute the orbifold Euler characteristic of $\mathcal{M}_g^n$ via techniques from Harer and Zagier, including generating functions and integral representations.
• Wick’s lemma is applied to relate generating series of $\widehat{f}$ to Gaussian integrals, enabling the computation of exponential generating functions for Euler characteristics.
• The actual Euler characteristic is derived from the orbifold Euler characteristic using a formula of Serre and Brown, which accounts for group actions and orbifold singularities.
• The method involves expanding generating series in terms of $\lambda^{2g-2}$, with coefficients tied to the Euler characteristics of strata and their automorphism groups.
• A correction is made to the generating function for $g=2$ by adding $D^5/4$, resolving an error in [2] and aligning the result with the authors' formulae.

Experimental results

Research questions

• RQ1 What is the orbifold Euler characteristic of $\overline{\mathcal{M}}_{g}^{n}$ for arbitrary $g$ and $n > 2 - 2g$?
• RQ2 What is the ordinary Euler characteristic of $\overline{\mathcal{M}}_{g}^{n}$, and how does it relate to the orbifold Euler characteristic?
• RQ3 Why do previous computations for $g=2$ disagree with the authors' results, and what correction resolves this discrepancy?
• RQ4 Can a unified generating function framework be constructed to compute Euler characteristics across all genera and puncture numbers?
• RQ5 How do the contributions of different stable graph types (e.g., loops, edges) to the Euler characteristic combine in the final formula?

Key findings

• The orbifold Euler characteristic of $\mathcal{M}_g^n$ is computed via generating functions and integral representations, with explicit formulae given in Theorem 3.2.
• The ordinary Euler characteristic of $\overline{\mathcal{M}}_g^n$ is derived from the orbifold Euler characteristic using Serre and Brown’s formula, as stated in Theorem 4.3.
• For $g=2$, the authors identify an error in [2] involving the coefficient of $D^5$, correcting it by adding $D^5/4$ to the generating function $K_2(t)$, which aligns the result with their formulae.
• The computed values for $e(\overline{\mathcal{M}}_g^n)$ for $g=2,3,4$ and $n=0$ to $6$ are provided in Table 2, confirming consistency with known results for $g=0,1$ and resolving prior inconsistencies.
• The generating function $\exp(\widehat{f})$ is shown to equal $\lim_{M\to\infty} \frac{1}{\sqrt{(2\pi)^M}} \int_{\mathbb{R}^M} \exp(Q(\lambda, y, \underline{x})) \, d\mu_M$, establishing a link to Gaussian integrals via Wick’s lemma.
• The final formula for the generating series of $e(\overline{\mathcal{M}}_g)$ is shown to satisfy $-\sum_{g\geq 2} e(\overline{\mathcal{M}}_g) \log(1 - \lambda^{2g-2}) = \sum_{g\geq 2} u_g \lambda^{2g-2}$, confirming the enumeration of disconnected coverings.