Automorphisms of tropical Hassett spaces

TL;DR

This paper determines the automorphism groups of tropical Hassett spaces, showing they are finite products of symmetric groups and relate to automorphisms of certain simplicial complexes, connecting tropical and algebraic geometry.

Contribution

It explicitly computes automorphism groups of tropical Hassett spaces for various weights and genus, linking them to automorphisms of associated simplicial complexes and algebraic Hassett spaces.

Findings

Automorphism groups are finite products of symmetric groups.

Automorphism groups are isomorphic to automorphisms of specific simplicial complexes.

Connection established between tropical and algebraic Hassett spaces via automorphisms.

Abstract

Given an integer $g \geq 0$ 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 $g \geq 1$ 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 $\{1, \ldots, n\}$ 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 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.