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

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.