[Paper Review] The slopes determined by n points in the plane

Jeremy L. Martin|ArXiv.org|Feb 10, 2003

Commutative Algebra and Its Applications9 references4 citations

TL;DR

This paper investigates the algebraic and combinatorial structure of the variety of slopes determined by $n$ points in the plane, showing that the ideal $I_n$ generated by tree polynomials of rigidity circuits in $K_n$ is prime, Cohen-Macaulay, and admits a Gröbner basis formed by wheel polynomials. The Hilbert series and degree of the slope variety are computed combinatorially via perfect matchings and binary total partitions, revealing deep connections to graph theory and Stanley-Reisner theory.

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 {\em 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 enumeration of trees.

Motivation & Objective

• To characterize the algebraic variety parametrizing all possible slope vectors from $n$ points in general position in the plane.
• To determine the defining ideal $I_n$ of the affine slope variety of the complete graph $K_n$.
• To establish that $I_n$ is prime, reduced, and Cohen-Macaulay, and to compute its Hilbert series and degree combinatorially.
• To relate the geometry of the slope variety to combinatorial invariants such as perfect matchings and binary total partitions.
• To prove that the tree polynomials of wheel subgraphs generate $I_n$ and form a Gröbner basis under a specific term order.

Proposed method

• Construct a monomial ideal $J_n$ from the initial terms of tree polynomials of wheel subgraphs under a graded lexicographic term order.
• Use Stanley-Reisner theory to associate a simplicial complex $\Delta(n)$ to $J_n$, whose faces correspond to squarefree monomials not in $J_n$.
• Establish a recursive structure for the facets of $\Delta(n)$, showing they are in bijection with binary total partitions, which are counted by double factorials.
• Prove shellability of $\Delta(n)$, which implies that $J_n$ and hence $I_n$ are Cohen-Macaulay.
• Use the Hilbert series formula derived from the $h$-vector of $\Delta(n)$, where $h(n,k)$ counts perfect matchings on $[1,2n-4]$ with exactly $k$ long pairs.
• Leverage the equality of codimension and degree between $J_n$ and $\sqrt{I_n}$, and the fact that $J_n$ is a Gröbner degeneration of $I_n$, to conclude that $I_n$ is prime and scheme-theoretically defines the slope variety.

Experimental results

Research questions

• RQ1 What is the defining ideal $I_n$ of the affine slope variety of $K_n$, and is it prime?
• RQ2 Can the Hilbert series of the slope variety be computed combinatorially, and what does it count?
• RQ3 Is the ideal $I_n$ generated by the tree polynomials of wheel subgraphs, and do they form a Gröbner basis?
• RQ4 What is the degree of the slope variety $\tilde{\mathcal{S}}(K_n)$, and how is it related to combinatorial objects like perfect matchings?
• RQ5 Is the Stanley-Reisner complex of $I_n$ shellable, and does this imply that the slope variety is Cohen-Macaulay?

Key findings

• The ideal $I_n$ defining the affine slope variety $\tilde{\mathcal{S}}(K_n)$ is prime and generated by the tree polynomials of all wheel subgraphs of $K_n$.
• The tree polynomials of wheel subgraphs form a Gröbner basis for $I_n$ under a specific graded lexicographic order.
• The degree of $\tilde{\mathcal{S}}(K_n)$ is $\frac{(2n-4)!}{2^{n-2}(n-2)!}$, equal to the number of perfect matchings on $[1,2n-4]$.
• The Hilbert series of $R_n/I_n$ is $\frac{\sum_{k=0}^{n-2} h(n,k) t^k}{(1-t)^{2n-3}}$, where $h(n,k)$ counts perfect matchings with exactly $k$ long pairs.
• The Stanley-Reisner complex $\Delta(n)$ is shellable, hence $R_n/I_n$ and $\tilde{\mathcal{S}}(K_n)$ are Cohen-Macaulay.
• The variety $\tilde{\mathcal{S}}(K_n)$ has dimension $2n-3$, consistent with the expected dimension of the configuration space of $n$ points in the plane modulo affine transformations.