Abstract The moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.

中文翻译：

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加权稳定曲线热带模量的拓扑

摘要 体积1的热带w加权稳定曲线的模空间Δg,w自然地等同于w加权稳定曲线的哈塞特空间中奇异曲线的除数的对偶复数。如果至少有两个权重为 1，我们证明 Δ0, w 与可能具有不同维度的球体的楔形和同伦。在权重向量的附加自然假设下，我们为空间的 Betti 数建立了明确的公式。我们展示了无限的权重族，其中空间 Δ0,w 是断开的，并且 Δ0,w 的基本群具有扭转。在后一种情况下，通用封面被证明具有自然的模块化解释。这使得空间的加权变体与 Vogtmann 和 Cavalieri-Hampe-Markwig-Ranganathan 先前研究的重/轻案例形成鲜明对比。