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The Toric Geometry of Triangulated Polygons in Euclidean Space

Published online by Cambridge University Press:  20 November 2018

Benjamin Howard ,
Christopher Manon  and
John Millson
Benjamin Howard
Affiliation:
Center for Communications Research, Princeton, NJ 08540, U.S.A. e-mail: bhoward73@gmail.com
Christopher Manon
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A. e-mail: manonc@math.umd.edu jjm@math.umd.edu
John Millson
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A. e-mail: manonc@math.umd.edu jjm@math.umd.edu
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Abstract

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Speyer and Sturmfels associated Gröbner toric degenerations $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{\mathcal{T}}}$ of $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{{}}}$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{r}^{\mathcal{T}}$ of ${{M}_{r}}$, 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.

Keywords

14L2453D20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 4 , 01 August 2011 , pp. 878 - 937
Copyright
Copyright © Canadian Mathematical Society 2011

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