与斜多项式代数及曲面三角剖分相关的超图
Hypergraphs associated to skew polynomial algebras and surface triangulations
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中文总结 AI 辅助
本文通过将斜多项式代数的点概形实现问题转化为3-一致超图（点超图）的组合研究，给出同调判据，构造仿射模空间簇并确定维数，证明点超图不能由有限禁止子图刻画，并完成六顶点情形的局部刻画。
中文摘要 AI 辅助
我们研究了斜多项式代数的点概形的实现问题。我们通过将其转化为对某些$3$-一致超图（称为点超图）的研究，以组合方式处理该问题。首先，我们给出了刻画点超图的同调判据。利用该判据，我们研究了从连通可定向闭曲面的三角剖分中删除三角形所得的超图。恰好删除一个三角形总是产生非点超图。当三角剖分没有分离的非面$3$-圈时，删除零个或至少两个三角形产生点超图，而单三角形删除是关于取诱导子超图而言的极小非点超图。由此可知，点超图不能由有限多个禁止诱导子超图来刻画。接下来，对于每个点超图，我们构造了一个实现它的斜多项式代数的仿射模空间簇，并确定其维数。对于由上述曲面构造产生的点超图，我们得到了一个用顶点数、欧拉示性数和删除三角形数表示的显式维数公式。最后，我们用一个四顶点局部条件以及一个单一例外障碍来刻画六顶点上的点超图。
英文摘要
We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.
发表机构
- Osaka University（大阪大学）
- Shinshu University（信州大学）
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