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15 January 2006 The slopes determined by n points in the plane
Jeremy L. Martin
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Duke Math. J. 131(1): 119-165 (15 January 2006). DOI: 10.1215/S0012-7094-05-13114-3

Abstract

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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Jeremy L. Martin. "The slopes determined by n points in the plane." Duke Math. J. 131 (1) 119 - 165, 15 January 2006. https://doi.org/10.1215/S0012-7094-05-13114-3

Information

Published: 15 January 2006
First available in Project Euclid: 15 December 2005

zbMATH: 1093.05018
MathSciNet: MR2219238
Digital Object Identifier: 10.1215/S0012-7094-05-13114-3

Subjects:
Primary: 05C10, 13P10, 14N20

Rights: Copyright © 2006 Duke University Press

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Vol.131 • No. 1 • 15 January 2006
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