The Toric Geometry of Triangulated Polygons in Euclidean Space

Details The Toric Geometry of Triangulated Polygons in Euclidean Space The Toric Geometry of Triangulated Polygons in Euclidean Space

2008-10-08

arxiv.org/abs/0810.1352v1

Mathematics

Symplectic Geometry

41 pages, 10 figures

Scientific paper

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].

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