Duke Mathematical Journal
- Duke Math. J.
- Volume 131, Number 1 (2006), 119-165.
The slopes determined by points in the plane
Let , , …, be the slopes of the () lines connecting points in general position in the plane. The ideal of all algebraic relations among the defines a configuration space called the slope variety of the complete graph. We prove that 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
Duke Math. J., Volume 131, Number 1 (2006), 119-165.
First available in Project Euclid: 15 December 2005
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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