Symmetries of tropical moduli spaces of curves

Kannan, Siddarth

doi:10.1090/tran/8393

Symmetries of tropical moduli spaces of curves
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by Siddarth Kannan PDF

Trans. Amer. Math. Soc. 374 (2021), 5805-5847 Request permission

Abstract:

Put $\Delta _{g, n} \subset M_{g, n}^\mathrm {trop}$ for the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. We compute the automorphism group $Aut(\Delta _{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$. In particular, we show that $Aut(\Delta _{g})$ is trivial for $g \geq 2$, while $Aut(\Delta _{g, n}) \cong S_n$ when $n \geq 1$ and $(g, n) \neq (0, 4), (1, 2)$. The space $\Delta _{g, n}$ is a symmetric $\Delta$-complex in the sense of Chan, Galatius, and Payne, and is identified with the dual intersection complex of the boundary divisor in the Deligne-Mumford-Knudsen moduli stack $\overline {\mathcal {M}}_{g, n}$ of stable curves. After the work of Massarenti [J. Lond. Math. Soc. 89 (2014), pp. 131–150], who has shown that $Aut(\overline {\mathcal {M}}_g)$ is trivial for $g \geq 2$ while $Aut(\overline {\mathcal {M}}_{g, n}) \cong S_n$ when $n \geq 1$ and $2g - 2 + n \geq 3$, our result implies that the tropical moduli space $\Delta _{g, n}$ faithfully reflects the symmetries of the algebraic moduli space for general $g$ and $n$.

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