欧几里得空间中三角剖分多边形的复曲面几何
The Toric Geometry of Triangulated Polygons in Euclidean Space
摘要：
Speyer和Sturmfels [SpSt]将$\mathrm{Gr}_2(\C^n)$的Gröbner复曲面退化$\mathrm{Gr}_2(\C^n)^{\tree}$与具有$n$叶的每个三价树$\tree$相关联。这些退化引起$M_{\br}$的复曲面退化$M_{\br}^{\tree}$，$n$空间在投影线上有序加权（由$\br$）点。本文的目的是给出复曲面纤维作为分层辛空间的几何（欧几里德多边形）描述，并将环面的紧部分的作用描述为“多边形的弯曲”。“我们证明了Foth和Hu [FH]的猜想，即复曲面纤维同胚于Kamiyama和Yoshida [KY]定义的空间。
英文摘要：
Speyer and Sturmfels [SpSt] associated Gröbner 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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