Duke Mathematical Journal

Geometry of Chow quotients of Grassmannians

Sean Keel and Jenia Tevelev

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We consider Kapranov's Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in Pr1 in linear general position. For r=2, this is canonically identified with the Grothendieck-Knudsen compactification of M0,n which has, among others, the following nice properties:

(1) modular meaning: stable pointed rational curves;

(2) canonical description of limits of one-parameter degenerations;

(3) natural Mori theoretic meaning: log-canonical compactification.

We generalize (1) and (2) to all (r,n), but we show that (3), which we view as the deepest, fails except possibly in the cases (2,n), (3,6), (3,7), (3,8), where we conjecture that it holds

Article information

Duke Math. J., Volume 134, Number 2 (2006), 259-311.

First available in Project Euclid: 8 August 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E
Secondary: 14D 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]


Keel, Sean; Tevelev, Jenia. Geometry of Chow quotients of Grassmannians. Duke Math. J. 134 (2006), no. 2, 259--311. doi:10.1215/S0012-7094-06-13422-1. https://projecteuclid.org/euclid.dmj/1155045503

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  • V. Alexeev, Log canonical singularities and complete moduli of stable pairs, preprint.
  • D. Barlet, ``Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension Finie'' in Fonctions de plusieurs variables complexes, II, Lecture Notes in Math. 482, Springer, Berlin, 1975, 1--158.
  • K. S. Brown, Buildings, reprint of 1989 original, Springer Monogr. Math., Springer, New York, 1998.
  • M. Cailotto, Algebraic connections on logarithmic schemes, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 1089--1094.
  • I. V. Dolgachev, ``Abstract configurations in algebraic geometry'' in The Fano Conference (Torino, 2002), Univ. Torino, Turin, 2004, 423--462.
  • I. V. Dolgachev and Y. Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5--56.
  • V. G. Drinfeld, Elliptic modules (in Russian), Mat. Sb. (N.S.) 94(136) (1974), 594--627., 656; English translation in Math. USSR-Sb. 23 (1974), 561--592.
  • G. Faltings, ``Toroidal resolutions for some matrix singularities'' in Moduli of Abelian Varieties (Texel Island, Netherlands, 1999), Prog. Math. 195, Birkhäuser, Basel, 2001, 157--184.
  • R. Friedman, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), 75--114.
  • W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
  • I. M. Gelfand, R. M. Goresky, R. D. Macpherson, and V. V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math. 63 (1987), 301--316.
  • I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Math. Theory Appl., Birkhäuser, Boston, 1994.
  • P. Hacking, Compact moduli of hyperplane arrangements, preprint.
  • P. Hacking, S. Keel, and J. Tevelev Compactification of the moduli space of hyperplane arrangements, to appear in J. Algebraic Geom.
  • D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, New York, 1952.
  • M. M. Kapranov, ``Chow quotients of Grassmannians, I'' in I. M. Gelfand Seminar, Adv. Soviet Math. 16, Part 2, Amer. Math. Soc., Providence, 1993, 29--110.
  • —, Veronese curves and Grothendieck-Knudsen moduli space $\oM_0,n$, J. Algebraic Geom. 2 (1993), 239--262.
  • M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), 643--655.
  • Y. Kawamata, K. Matsuda, and K. Matsuki, ``Introduction to the minimal model program'' in Algebraic Geometry (Sendai, Japan, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 283--360.
  • Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothings of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395--409.
  • S. Keel and J. Mckernan, Contractible extremal rays of $\overlineM_0,n$, preprint.
  • J. KolláR (with 14 coauthors), Flips and Abundance for Algebraic Threefolds (Salt Lake City, 1991), Astérique 211, Soc. Math. France, Montrouge, 1992.
  • L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et compactification des $\rm PGL_r^\it n+1/\rm PGL_r$, Invent. Math. 136 (1999), 233--271.
  • —, Chirurgie des grassmanniennes, CRM Monogr. Ser. 19, Amer. Math. Soc., Providence, 2003.
  • G. A. Mustafin, Non-Archimedean uniformization, Math. USSR-Sb. 34 (1978), 187--214.
  • T. Oda, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin, 1988.
  • M. C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), 747, --791.
  • —, The logarithmic cotangent complex, Math. Ann. 333 (2005), 859--931.
  • E. H. Spanier, Algebraic Topology, corrected reprint of 1966 original, Springer, New York, 1981.
  • D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), 389--411.