Automorphisms of tropical Hassett spaces

Abstract

Given an integer $g\ge 0$ and a weight vector $w\in {\mathbb{Q}}^n\cap (0,1{]}^n$ satisfying $2g-2+\sum w_i>0$, let ${\Delta }_{g,w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $\mathrm{Aut}({\Delta }_{g,w})$ for $g\ge 1$ and arbitrary $w$, and we calculate the group $\mathrm{Aut}({\Delta }_{0,w})$ when $w$ is heavy/light. In both of these cases, we show that $\mathrm{Aut}({\Delta }_{g,w})\cong \mathrm{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on $\{1,\dots ,n\}$ whose faces are subsets with $w$-weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space ${\Delta }_{g,w}$ may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space ${\overline{\mathcal{M}}}_{g,w}$. Following the work of Massarenti and Mella (2017) on the biregular automorphism group $\mathrm{Aut}({\overline{\mathcal{M}}}_{g,w})$, we show that $\mathrm{Aut}({\Delta }_{g,w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.