In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$ . We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$ , where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$ . This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $ .

中文翻译：

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复曲面中哈纳克曲线的模空间

2006 年，Kenyon 和 Okounkov Kenyon 和 Okounkov [12] 计算了 Harnack 度数曲线的模空间d在 ${\mathbb {C}\mathbb {P}}^2$ . 我们将它们的构造推广到任何投影复曲面并证明模空间 ${\数学{H}_\Delta }$ 具有牛顿多边形的哈纳克曲线 $\三角洲$ 微分同胚于 ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+gm}$ ， 在哪里 $\三角洲$ 已米边缘，G内格点和n边界格点。这解决了 Crétois 和 Lang 的猜想。主要结果使用抽象的热带曲线来构建这个模空间的紧凑化，其中额外的点对应于曲线的集合，这些曲线可以拼凑在一起以产生曲线 ${\数学{H}_\Delta }$ . 这种紧化有一个自然分层，与第二多面体的偏序相同 $\三角洲$ .