The Toric Geometry of Triangulated Polygons in Euclidean Spac

摘要

Speyer and Sturmfels associated Gr?bner toricdegenerations $mathrm{Gr}_2(mathbb{C}^n)^{mathcal{T}}$ of $mathrm{Gr}_2(mathbb{C}^n)$ with eachtrivalent tree $mathcal{T}$ having $n$ leaves. These degenerationsinduce toricdegenerations $M_{mathbf{r}}^{mathcal{T}}$ of $M_{mathbf{r}}$, thespace of $n$ ordered, weighted (by $mathbf{r}$) points on the projective line. Our goal in this paper is to give ageometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torusas "bendings of polygons". We prove the conjecture of Foth and Hu thatthe toric fibers are homeomorphicto the spaces defined by Kamiyama and Yoshida.