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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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.

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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