Moduli spaces of metric graphs of genus 1 with marks on vertices
Dmitry N. Kozlov

TL;DR
This paper demonstrates that certain moduli spaces of genus 1 metric graphs with vertex marks are homotopy equivalent to tropical curve moduli spaces, using explicit homotopies and the scanning homotopy technique.
Contribution
The paper establishes a homotopy equivalence between moduli spaces of genus 1 metric graphs with vertex marks and tropical curve moduli spaces, introducing explicit homotopies.
Findings
Homotopy equivalence between $MG_{1,n}^v$ and $TM_{1,n}$
Use of explicit homotopies and scanning homotopy
Conjecture on generalization to higher genus
Abstract
In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with marks on vertices are homotopy equivalent to the spaces , which are the moduli spaces of tropical curves of genus 1 with marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.
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Taxonomy
TopicsPolynomial and algebraic computation · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications
