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.


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


Zbl 1029.05040


Macaulay2; OEIS
Full Text: DOI arXiv

Online Encyclopedia of Integer Sequences:

Number of ways of arranging n lines in the (affine) plane.


[1] D. Bayer and M. Stillman, Macaulay: A computer algebra system for algebraic geometry, 1994, http://www.math.columbia.edu/\(\sim\)bayer/Macaulay/
[2] L. J. Billera, S. P. Holmes, and K. Vogtmann, Geometry of the space of phylogenetic trees , Adv. in Appl. Math. 27 (2001), 733–767. · Zbl 0995.92035 · doi:10.1006/aama.2001.0759
[3] W. Bruns and J. Herzog, Cohen-Macaulay Rings , Cambridge Stud. Adv. Math. 39 , Cambridge Univ. Press, Cambridge, 1993.
[4] C. De Concini and C. Procesi, Wonderful models of subspace arrangements , Selecta Math. (N.S.) 1 (1995), 459–494. · Zbl 0842.14038 · doi:10.1007/BF01589496
[5] D. Eisenbud, Commutative Algebra: With a View toward Algebraic Geometry , Grad. Texts in Math. 150 , Springer, New York, 1995. · Zbl 0819.13001
[6] W. Fulton and R. Macpherson, A compactification of configuration spaces , Ann. of Math. (2) 139 (1994), 183–225. · Zbl 0820.14037 · doi:10.2307/2946631
[7] J. Graver, B. Servatius, and H. Servatius, Combinatorial Rigidity , Grad. Stud. Math. 2 , Amer. Math. Soc., Providence, 1993. · Zbl 0788.05001
[8] G. Kreweras and Y. Poupard, Sur les partitions en paires d’un ensemble fini totalement ordonné , Publ. Inst. Statist. Univ. Paris 23 (1978), 57–74. · Zbl 0423.05007
[9] J. L. Martin, Geometry of graph varieties , Trans. Amer. Math. Soc. 355 (2003), 4151–4169. JSTOR: · Zbl 1029.05040 · doi:10.1090/S0002-9947-03-03321-X
[10] -, Graph varieties , Ph.D. dissertation, University of California, San Diego, La Jolla, Calif., 2002, http://www.math.ku.edu/\(\sim\)jmartin/pubs.html N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2003, updated 2005, http://www.research.att.com/\(\sim\)njas/sequences/
[11] R. P. Stanley, Combinatorics and Commutative Algebra , 2nd ed., Progr. Math. 41 , Birkhäuser, Boston, 1996. · Zbl 0838.13008
[12] -, Enumerative Combinatorics , Vol. 2, Cambridge Stud. Adv. Math. 62 , Cambridge Univ. Press, Cambridge, 1999.
[13] W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry , Algorithms Comput. Math. 2 , Springer, Berlin, 1998. · Zbl 0896.13021
[14] D. B. West, Introduction to Graph Theory , 2nd ed., Prentice Hall, Upper Saddle River, N.J., 2001. · Zbl 0992.83079
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