[Paper Review] On the S_n-equivariant Euler characteristic of M_{2,n}

Eugene Gorsky|ArXiv.org|Jul 18, 2007

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper derives the $S_n$-equivariant Euler characteristic of the moduli space $\mathcal{M}_{2,n}$ of genus 2 curves with $n$ marked points by extending Getzler's formula to varieties with nontrivial automorphism groups. It accounts for the action of the automorphism group on configuration spaces of points, using group actions and Adams operations to compute the generating function for the equivariant Euler characteristic, yielding a closed-form formula for all $n$. The key result is a precise generating function that incorporates orbifold contributions from curves with different symmetry types.

ABSTRACT

The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\mathcal{M}_{2,n}$.

Motivation & Objective

• To compute the $S_n$-equivariant Euler characteristic of the moduli space $\mathcal{M}_{2,n}$ of genus 2 curves with $n$ marked points.
• To extend Getzler's formula for $S_n$-equivariant Hodge-Deligne polynomials to varieties with nontrivial automorphism groups.
• To account for the action of $\mathrm{Aut}(C)$ on configuration spaces $F(C,n)$ in the fiber of the forgetful map $\mathcal{M}_{2,n} \to \mathcal{M}_2$.
• To derive a generating function for the $S_n$-equivariant Euler characteristic by analyzing strata in $\mathcal{M}_2$ with different automorphism groups.

Proposed method

• Adapt Getzler's formula to include group actions by incorporating the Grothendieck ring of representations of finite groups and using Adams operations.
• Use the Lefshetz fixed-point theorem to compute characters of group actions on cohomology, relating them to fixed-point counts.
• Compute the $S_n$-equivariant Euler characteristic of $F(X,n)/G$ for a finite group $G$ acting on $X$, using a generating function involving Newton polynomials $p_k$ and orbit data.
• Classify strata in $\mathcal{M}_2$ by automorphism group type (e.g., hyperelliptic involution, larger groups), and compute their contributions via orbit counting and orbifold Euler characteristics.
• Apply the formula to $\mathcal{M}_{2,n}$ by analyzing fibers over curves with different automorphism groups, particularly $\mathbb{CP}^1$ quotients under group actions.
• Derive a generating function for the $S_n$-equivariant Euler characteristic using the structure of symmetric configurations on $\mathbb{CP}^1$ and known results on hyperelliptic moduli spaces.

Experimental results

Research questions

• RQ1 How can Getzler's formula for $S_n$-equivariant Hodge-Deligne polynomials be generalized to varieties with nontrivial automorphism groups?
• RQ2 What is the $S_n$-equivariant Euler characteristic of $\mathcal{M}_{2,n}$, the moduli space of genus 2 curves with $n$ marked points?
• RQ3 How do automorphism groups of genus 2 curves affect the equivariant topology of configuration spaces of marked points?
• RQ4 Can the generating function for the $S_n$-equivariant Euler characteristic of $\mathcal{M}_{2,n}$ be expressed in closed form using group action data and orbifold corrections?
• RQ5 What is the role of the hyperelliptic involution and other automorphisms in computing the orbifold Euler characteristic of $\mathcal{M}_{2,n}$?

Key findings

• The $S_n$-equivariant Euler characteristic of $\mathcal{M}_{2,n}$ is derived via a generalized formula that accounts for automorphism group actions on point configurations.
• The generating function for the $S_n$-equivariant Euler characteristic is expressed as a rational function involving Newton polynomials $p_k$ and orbit data from group actions.
• For genus 2, the fiber of the forgetful map $\mathcal{M}_{2,n} \to \mathcal{M}_2$ over a curve $C$ is $F(C,n)/\mathrm{Aut}(C)$, not $F(C,n)$, and this quotient is isomorphic to $\mathbb{CP}^1$ under the action of $\mathrm{Aut}(C)$.
• The orbifold Euler characteristic of $\mathcal{H}_{g,0}$ is $-{1\over 4g(2g+1)(2g+2)}$, which contributes to the generating function for hyperelliptic moduli spaces.
• For $g=2$, the Euler characteristic $\chi(\mathcal{H}_{2,5})$ vanishes, a result that is new and nontrivial.
• The generating function for $\chi(\mathcal{H}_{g,n})$ is given by a rational expression involving $a$, $b$, $c$, and $d$, with $d = -{1\over 4g(2g+1)(2g+2)}$, and satisfies a linear constraint $2a + 2b + c = 1 + {1\over 2g(2g+1)(2g+2)}$.