热带Hassett空间的自同构
Automorphisms of tropical Hassett spaces
AI总结：
本文计算了热带Hassett空间 $\Delta_{g, w}$ 的自同构群，证明其同构于相关单纯复形 $K_w$ 的自同构群，即对称群的有限直积，并揭示其与代数Hassett空间中保持奇异曲线除子的自同构子群的自然等同。
摘要：
给定整数 $g \geq 0$ 和满足 $2g - 2 + \sum w_i > 0$ 的权向量 $w \in \mathbb{Q}^n \cap (0, 1]^n$，令 $\Delta_{g, w}$ 表示亏格为 $g$、体积为一的 $n$ 个标记的 $w$-稳定热带曲线的模空间。我们计算了 $g \geq 1$ 且任意 $w$ 时的自同构群 $\mathrm{Aut}(\Delta_{g, w})$，并在 $w$ 为重/轻时计算了群 $\mathrm{Aut}(\Delta_{0, w})$。在这两种情况下，我们证明 $\mathrm{Aut}(\Delta_{g, w}) \cong \mathrm{Aut}(K_w)$，其中 $K_w$ 是定义在集合 $\{1, \ldots, n\}$ 上的抽象单纯复形，其面是 $w$-权至多为 $1$ 的子集。我们证明这些群恰好是对称群的有限直积。空间 $\Delta_{g, w}$ 也可以等同于代数Hassett空间 $\overline{\mathcal{M}}_{g, w}$ 中奇异曲线除子的对偶复形。继Massarenti和Mella关于双正则自同构群 $\mathrm{Aut}(\overline{\mathcal{M}}_{g, w})$ 的工作之后，我们证明 $\mathrm{Aut}(\Delta_{g, w})$ 自然等同于保持奇异曲线除子的自同构子群。
英文摘要：
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 $Δ_{g, w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $\mathrm{Aut}(Δ_{g, w})$ for $g \geq 1$ and arbitrary $w$, and we calculate the group $\mathrm{Aut}(Δ_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $\mathrm{Aut}(Δ_{g, w}) \cong \mathrm{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on $\{1, ..., n\}$ 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 $Δ_{g, w}$ 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 $\mathrm{Aut}(Δ_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.
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