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The Toric Geometry of Triangulated Polygons in Euclidean Spac

  Published:2011-04-14
 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.
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Abstract

Speyer and Sturmfels associated Gröbner 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.
MSC Classifications: 14L24, 53D20 show english descriptions Geometric invariant theory [See also 13A50]
Momentum maps; symplectic reduction
14L24 - Geometric invariant theory [See also 13A50]
53D20 - Momentum maps; symplectic reduction
 

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