Computing the Chow Variety of Quadratic Space Curves

  • Peter Bürgisser
  • Kathlén KohnEmail author
  • Pierre Lairez
  • Bernd Sturmfels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional projective space. We study the subvariety of coisotropic hypersurfaces. Following Gel’fand, Kapranov and Zelevinsky, it decomposes into Chow forms of plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric surfaces. We compute the ideals of these loci.


Chow variety Coisotropic hypersurface Grassmannian Space curve Computation 


  1. 1.
    Bürgisser, P., Cucker, F.: Condition: The Geometry of Numerical Algorithms. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Catanese, F.: Cayley forms and self-dual varieties. Proc. Edinb. Math. Soc. 57, 89–109 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cayley, A.: On the theory of elimination. Camb. Dublin Math. J. 3, 116–120 (1848)Google Scholar
  4. 4.
    Chow, W.-L., van der Waerden, B.L.: Zur algebraischen Geometrie. IX. Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten. Math. Ann. 113, 696–708 (1937)CrossRefzbMATHGoogle Scholar
  5. 5.
    DeConcini, C., Goresky, M., MacPherson, R., Procesi, C.: On the geometry of quadrics and their degenerations. Comment. Math. Helvetici 63, 337–413 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry.
  8. 8.
    Green, M., Morrison, I.: The equations defining Chow varieties. Duke Math. J. 53, 733–747 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sturmfels, B.: The Hurwitz form of a projective variety. arXiv:1410.6703

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Kathlén Kohn
    • 1
    Email author
  • Pierre Lairez
    • 1
  • Bernd Sturmfels
    • 1
    • 2
  1. 1.Institute of MathematicsTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations