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.

v2: updated proof of lemma on torus localisation and exposition. Version accepted at Crelle