The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

TL;DR

This paper computes the $S_n$-equivariant Euler characteristic of the moduli space of genus one stable maps to projective space, using torus actions, symmetric functions, and graph coloring enumeration to derive explicit formulas.

Contribution

It introduces a new method linking torus actions, symmetric functions, and graph colorings to compute equivariant Euler characteristics of Kontsevich moduli spaces.

Findings

Derived a closed formula for the Euler characteristic of fixed point loci under $ ext{C}^*$-action.

Expressed the motive of the moduli space in terms of non-rational tail subspaces and plethysm.

Connected geometric, combinatorial, and symmetric function techniques to compute topological invariants.

Abstract

We compute the $S_n$-equivariant topological Euler characteristic of the Kontsevich moduli space $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$. Letting $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d) \subset \overline{\mathcal{M}}_{1, n}(\P^r, d)$ denote the subspace of maps from curves without rational tails, we solve for the motive of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$ in terms of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)$ and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic $\mathbb{C}^\star$-action on $\mathbb{P}^r$, we derive a closed formula for the Euler characteristic of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)^{\mathbb{C}^\star}$ as an $S_n$-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types $A$ and $B$, as well as the enumeration of graph colourings with prescribed symmetry.