The slopes determined by n points in the plane

Details The slopes determined by n points in the plane The slopes determined by n points in the plane

2003-02-10

arxiv.org/abs/math/0302106v3

Duke Math. J. 131, no. 1 (2006), 119-165

Mathematics

Algebraic Geometry

36 pages; final published version

Scientific paper

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 Gr\"obner 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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