Duke Mathematical Journal

The equations for the moduli space of n points on the line

Benjamin Howard, John Millson, Andrew Snowden, and Ravi Vakil

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory (GIT) quotients come with a natural ample line bundle and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold, the space of six points with equal weight. We also show that the ideal of relations is generated in degree at most four, and we give an explicit description of the generators. If all the weights are even (e.g., in the case of equal weight for odd n), then we show that the ideal of relations is generated by quadrics

Article information

Source
Duke Math. J., Volume 146, Number 2 (2009), 175-226.

Dates
First available in Project Euclid: 5 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1231170938

Digital Object Identifier
doi:10.1215/00127094-2008-063

Mathematical Reviews number (MathSciNet)
MR2477759

Zentralblatt MATH identifier
1161.14033

Subjects
Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14D22: Fine and coarse moduli spaces 14H10: Families, moduli (algebraic)

Citation

Howard, Benjamin; Millson, John; Snowden, Andrew; Vakil, Ravi. The equations for the moduli space of $n$ points on the line. Duke Math. J. 146 (2009), no. 2, 175--226. doi:10.1215/00127094-2008-063. https://projecteuclid.org/euclid.dmj/1231170938


Export citation

References

  • V. V. Batyrev and O. N. Popov, ``The Cox ring of a del Pezzo surface'' in Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, Calif., 2002), Progr. Math. 226, Birkhäuser, Boston, 2004, 85--103.
  • W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997), 235--265.
  • N. Bourbaki, Commutative Algebra: Chapters 1--7, Elem. Math. (Berlin), Springer, Berlin, 1998.
  • J. A. De Loera and T. B. Mcallister, Vertices of Gelfand-Tsetlin polytopes, Discrete Comput. Geom. 32 (2004), 459--470.
  • H. Derksen, Universal denominators of Hilbert series, J. Algebra 285 (2005), 586--607.
  • I. Dolgachev, Lectures on Invariant Theory, London Math. Soc. Lecture Note Ser. 296, Cambridge Univ. Press, Cambridge, 2003. ?
  • I. Dolgachev and D. Ortland, Point Sets in Projective Spaces and Theta Functions, Astérisque 165, Soc. Math. France, Montrouge, 1988.
  • P. Foth and Y. Hu, ``Toric degenerations of weight varieties and applications'' in Travaux mathématiques, fasc. 16 (Luxembourg, 2004), Trav. Math. 16, Univ. Luxemb., Luxembourg, 2005, 87--105.
  • N. Gonciulea and V. Lakshmibai, Flag Varieties, Travaux en Cours 63, Hermann, Paris, 2001.
  • J.-C. Hausmann and A. Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), 173--198.
  • B. Howard, C. Manon, and J. Millson, The toric geometry of polygons in Euclidean space, in preparation.
  • B. Howard, J. Millson, A. Snowden, and R. Vakil, computer code for this article, http://math.stanford.edu/$\sim$vakil/HMSV/
  • Y. Hu, Stable configurations of linear subspaces and quotient coherent sheaves, Q. J. Pure Appl. Math. 1 (2005), 127--164.
  • Y. Kamiyama and T. Yoshida, Symplectic toric space associated to triangle inequalities, Geom. Dedicata 93 (2002), 25--36.
  • M. Kapovich and J. J. Millson, The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), 479--513.
  • S. Keel and J. Tevelev, Equations for $\overline M_0,n$, preprint,\arxivmath/0507093v1[math.AG]
  • A. Kempe, On regular difference terms, Proc. London Math. Soc. 25 (1894), 343--359.
  • D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • P. J. Olver and C. Shakiban, Graph theory and classical invariant theory, Adv. Math. 75 (1989), 212--245.
  • T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967), 1022--1046.
  • N. J. A. Sloane, The on-line encyclopedia of integer sequences, available at http://www.research.att.com/$\sim$njas/sequences/
  • D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), 389--411.
  • B. Sturmfels, Algorithms in Invariant Theory, Texts Monogr. Symbol. Comput., Springer, Vienna, 1993.
  • —, Gröbner Bases and Convex Polytopes, Univ. Lecture Ser. 8, Amer. Math. Soc., Providence, 1996.