The Toric Geometry of Triangulated Polygons in Euclidean Space

TL;DR

This paper provides a geometric description of toric degenerations of moduli spaces of weighted points on the projective line, relating them to Euclidean polygons and confirming a conjecture about their topological structure.

Contribution

It offers a Euclidean polygon interpretation of toric fibers in moduli spaces and proves a conjecture linking these fibers to known topological spaces.

Findings

Toric fibers are stratified symplectic spaces.

The action of the torus corresponds to polygon bendings.

Confirmed the conjecture that fibers are homeomorphic to Kamiyama and Yoshida's spaces.

Abstract

Speyer and Sturmfels [SpSt] associated Gr\"obner toric degenerations $\mathrm{Gr}_2(\C^n)^{\tree}$ of $\mathrm{Gr}_2(\C^n)$ to each trivalent tree $\tree$ with $n$ leaves. These degenerations induce toric degenerations $M_{\br}^{\tree}$ of $M_{\br}$, the space of $n$ ordered, weighted (by $\br$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons." We prove the conjecture of Foth and Hu [FH] that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida [KY].