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The Toric Geometry of Triangulated Polygons in Euclidean Spac |
Canad. J. Math. 63(2011), 878-937
Published:2011-04-14
Printed: Aug 2011
Printed: Aug 2011
- Benjamin Howard,
- Christopher Manon,
- John Millson,
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 - Geometric invariant theory [See also 13A50] 53D20 - Momentum maps; symplectic reduction |