由平面上n个点确定的斜率
The slopes determined by n points in the plane
摘要：
令$m_{12}$，$m_{13}$，.，$m_{n-1,n}$是连接平面中一般位置的$n$点的$\binom{n}{2}$线的斜率。$m_{ij}$之间所有代数关系的理想$I_n$定义了一个称为{完全图的\em斜率簇}的位形空间。证明了$I_n$是约化的，给出了它的Cohen-Macaulay显式Gröbner基，并组合地计算了它的Hilbert级数.我们主要通过研究相关的Stanley Reisner单纯复形来进行，它具有复杂的递归结构。此外，我们能够回答许多问题的几何形状的斜率品种，将它们转化为纯粹的组合问题有关枚举的树木。
英文摘要：
Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called the {\em slope variety of the complete graph}. We prove that $I_n$ is reduced and Cohen-Macaulay, give an explicit Gröbner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees.
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