Abstract view
The Toric Geometry of Triangulated Polygons in Euclidean Spac


Published:20110414
Printed: Aug 2011
Benjamin Howard,
Center for Communications Research, Princeton, NJ 08540, U.S.A.
Christopher Manon,
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
John Millson,
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
Abstract
Speyer and Sturmfels associated GrĂ¶bner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.