Duke Mathematical Journal

The slopes determined by n points in the plane

Jeremy L. Martin

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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

Article information

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

Dates
First available in Project Euclid: 15 December 2005

Permanent link to this document
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

Subjects
Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14N20: Configurations and arrangements of linear subspaces

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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References

  • D. Bayer and M. Stillman, Macaulay: A computer algebra system for algebraic geometry, 1994, http://www.math.columbia.edu/$\sim$bayer/Macaulay/
  • L. J. Billera, S. P. Holmes, and K. Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math. 27 (2001), 733--767.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.
  • C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), 459--494.
  • D. Eisenbud, Commutative Algebra: With a View toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
  • W. Fulton and R. Macpherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), 183--225.
  • J. Graver, B. Servatius, and H. Servatius, Combinatorial Rigidity, Grad. Stud. Math. 2, Amer. Math. Soc., Providence, 1993.
  • 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.
  • J. L. Martin, Geometry of graph varieties, Trans. Amer. Math. Soc. 355 (2003), 4151--4169.
  • —, 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/
  • R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston, 1996.
  • —, Enumerative Combinatorics, Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge Univ. Press, Cambridge, 1999.
  • W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algorithms Comput. Math. 2, Springer, Berlin, 1998.
  • D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River, N.J., 2001.