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# Topology of tropical moduli spaces of weighted stable curves in higher genus

@article{Kannan2020TopologyOT,
title={Topology of tropical moduli spaces of weighted stable curves in higher genus},
author={Siddarth Kannan and Shiyue Li and Stefano Serpente and Claudia He Yun},
journal={arXiv: Combinatorics},
year={2020}
}
Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $\Delta_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $\Delta_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $\Delta_{g, w}$ is simply connected for all values of $w… 2 Citations ## Figures and Tables from this paper Filtrations of moduli spaces of tropical weighted stable curves We study how changing the weight datum A = (a1, ..., an) ∈ (Q ∩ (0, 1]) n affects the topology of tropical moduli spaces M trop g,A , M trop g,A and ∆g,A and the homology of the latter one. We show Automorphisms of tropical Hassett spaces • Mathematics • 2021 Given an integer g ≥ 0 and a weight vector w ∈ Q∩(0, 1] satisfying 2g−2+ ∑ wi > 0, let ∆g,w denote the moduli space of n-marked, w-stable tropical curves of genus g and volume one. We calculate the ## References SHOWING 1-10 OF 36 REFERENCES Topology of the tropical moduli spaces$M_{2,n}$We study the topology of the link$M^{\mathrm{trop}}_{g,n}[1]$of the tropical moduli spaces of curves when g=2. Tropical moduli spaces can be identified with boundary complexes for Topology of moduli spaces of tropical curves with marked points In this paper we study topology of moduli spaces of tropical curves of genus$g$with$n$marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\it moduli Tropical curves, graph complexes, and top weight cohomology of ℳ_{ℊ} • Mathematics • 2018 We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY • Mathematics Forum of Mathematics, Sigma • 2016 We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector$w$of weights, the moduli space of tropical$w\$ -stable curves can be
Topology of tropical moduli of weighted stable curves
• Mathematics
• 2017
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
Tropical curves, graph homology, and top weight cohomology of M_g
• Mathematics
• 2018
We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight
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• Mathematics
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This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study
The virtual cohomological dimension of the mapping class group of an orientable surface
Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those
Tropical geometry of moduli spaces of weighted stable curves
A tropical analogue of Hassett's moduli spaces of weighted stable curves is defined and it is shown that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton.
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Abstract A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum