Duke Mathematical Journal

The slopes determined by n points in the plane

Jeremy L. Martin

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Let m12, m13, …, mn-1,n be the slopes of the (n2) lines connecting n points in general position in the plane. The ideal In of all algebraic relations among the mij defines a configuration space called the slope variety of the complete graph. We prove that In 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 the enumeration of trees

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Duke Math. J., Volume 131, Number 1 (2006), 119-165.

First available in Project Euclid: 15 December 2005

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Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14N20: Configurations and arrangements of linear subspaces


Martin, Jeremy L. The slopes determined by $n$ points in the plane. Duke Math. J. 131 (2006), no. 1, 119--165. doi:10.1215/S0012-7094-05-13114-3. https://projecteuclid.org/euclid.dmj/1134666123

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