JournalspmVol. 79, No. 1/2pp. 163–197

Automorphisms of tropical Hassett spaces

  • Sam Freedman

    Brown University, Providence, USA
  • Joseph Hlavinka

    Brown University, Providence, USA
  • Siddarth Kannan

    Brown University, Providence, USA
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Abstract

Given an integer g0g \geq 0 and a weight vector wQn(0,1]nw \in \mathbb{Q}^n \cap (0, 1]^n satisfying 2g2+wi>02g - 2 +\sum w_i > 0, let Δg,w\Delta_{g, w} denote the moduli space of nn-marked, ww-stable tropical curves of genus gg and volume one. We calculate the automorphism group Aut(Δg,w)\operatorname{Aut}(\Delta_{g, w}) for g1g \geq 1 and arbitrary ww, and we calculate the group Aut(Δ0,w)\operatorname{Aut}(\Delta_{0, w}) when ww is heavy/light. In both of these cases, we show that Aut(Δg,w)Aut(Kw)\operatorname{Aut}(\Delta_{g, w}) \cong \operatorname{Aut}(K_w), where KwK_w is the abstract simplicial complex on {1,,n}\{1, \ldots, n\} whose faces are subsets with ww-weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space Δg,w\Delta_{g, w} may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space Mg,w\overline{\mathcal{M}}_{g, w}. Following the work of Massarenti and Mella (2017) on the biregular automorphism group Aut(Mg,w)\operatorname{Aut}(\overline{\mathcal{M}}_{g, w}), we show that Aut(Δg,w)\operatorname{Aut}(\Delta_{g, w}) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

Cite this article

Sam Freedman, Joseph Hlavinka, Siddarth Kannan, Automorphisms of tropical Hassett spaces. Port. Math. 79 (2022), no. 1/2, pp. 163–197

DOI 10.4171/PM/2075