## Duke Mathematical Journal

### The slopes determined by $n$ points in the plane

Jeremy L. Martin

#### Abstract

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 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 the enumeration of trees

#### Article information

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

Dates
First available in Project Euclid: 15 December 2005

https://projecteuclid.org/euclid.dmj/1134666123

Digital Object Identifier
doi:10.1215/S0012-7094-05-13114-3

Mathematical Reviews number (MathSciNet)
MR2219238

Zentralblatt MATH identifier
1093.05018

#### Citation

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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