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The slopes determined by \(n\) points in the plane. (English) Zbl 1093.05018

Let \(p_1,\dots, p_n\) be \(n\) distinct points in the affine plane \(A^2_{\mathbf k}\) with no two points lying on the same vertical line. Let \(m_{ij}\in\mathbf k\) be the slope of the unique line joining \(p_i\) and \(p_j\). Thus \((m_{12},\dots,m_{n-1,n})\) is a point in affine space \(A^s_{\mathbf k}\) with \(s=\frac{n(n-1)}{2}\). The affine slope variety \(P(n)\) is defined as the closure of the locus of all such points arising from the data \((p_1,\dots,p_n)\). Note that in [Trans. Am. Math. Soc. 355, 4151–4169 (2003; Zbl 1029.05040)] the author considered the affine slope variety \(P(G)\) defined by a graph \(G\), and the variety \(P(n)=P(K_n)\) is defined by the complete graph \(K_n\) in these terms.
The affine slope variety turns out to have an unexpectedly rich combinatorial and geometric structure. In order to investigate its properties the author uses combinatorics (graph theory and recursive enumeration of trees), commutative algebra (Gröbner bases and Stanley-Reisner theory), and algebraic geometry.
Let us formulate the main results of the article. Let \(R_n=\mathbf k[m_{12},\dots,m_{n-1,n}]\) and \(I_n\) be the ideal generated by the tree polynomials of all rigidity circuits in the complete graph \(K_n\). Then the variety \(P(n)\) is defined scheme-theoretically by \(I_n\). Moreover, the tree polynomials of the so-called wheel subgraphs of \(K_n\) already generate \(I_n\) and form a Gröbner basis with respect to a certain graded lexicographic order. The dimension of \(P(n)\) equals \(2n-3\), and explicit formulas for the degree of \(P(n)\) and for the Hilbert series of \(R_n/I_n\) in terms of perfect matchings on the set \(\{1,2,\dots,2n-4\}\) are given. It is also shown that the ring \(R_n/I_n\) is Cohen-Macaulay.
Other spaces related to graph varieties include the Fulton-Macpherson compactification of configuration space and the De Concini-Procesi wonderful model of subspace arrangements. The results of this article may serve as a starting point for studying these relations in more detail.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14N20 Configurations and arrangements of linear subspaces

Citations:

Zbl 1029.05040

Software:

Macaulay2; OEIS
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References:

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