Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Invariants and projections of six lines in projective space

Author: Dana R. Vazzana
Journal: Trans. Amer. Math. Soc. 353 (2001), 2673-2688
MSC (2000): Primary 14L24; Secondary 14Q10
Published electronically: January 18, 2001
MathSciNet review: 1828467
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Given six lines in $\mathbf{P}^3$, quartics through the six lines define a map from $\mathbf{P}^3$ to $\mathbf{P}^4$, and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a ``double-sixer'' on a cubic surface.

References [Enhancements On Off] (What's this?)

  • [Ba] H. Baker.
    Principles of Geometry, Vol. 3.
    Cambridge Univ. Press, 1927.
  • [Ca] A. Cayley.
    On the surfaces each the locus of the vertex of a cone which passes through $m$ given points and touches $6-m$ given lines.
    Proc. London Math. Soc., 4:11-47, 1872.
  • [Co] A. Coble.
    Algebraic Geometry and Theta Functions.
    Amer. Math. Soc., 1929.
  • [DO] I. Dolgachev and D. Ortland.
    Point sets in projective space and theta functions.
    Asterisque, Volume 165. Soc. Math. France, 1989. MR 90i:14009
  • [D] I. Dolgachev.
    Introduction to geometric invariant theory.
    Lecture Notes Series, Number 25. Seoul National University, 1994. MR 96a:14019
  • [GH] P. Griffiths and J. Harris.
    Principles of Algebraic Geometry.
    Wiley, 1978. MR 80b:14001
  • [Gr] F. Grosshans, R. Gleeson, M. Hirsch, R. M. Williams.
    Object-image equations for x points and 6 - x lines.
    Preprint, 1998.
  • [Ha] R. Hartshorne.
    Algebraic Geometry.
    Springer-Verlag, 1977. MR 57:3116
  • [Hz] V. Hierhölzer.
    Ueber Kegelschnitte im Raume.
    Math. Ann., 2:563 -585, 1870.
  • [Hg] R. Huang.
    Combinatorial Methods in Invariant Theory.
    PhD thesis, Massachusetts Institute of Technology, 1993.
  • [Ht] B. Hunt.
    The Geometry of Some Special Arithmetic Quotients.
    Springer-Verlag, 1996. MR 98c:14033
  • [Ht2]
  • [K] M. Kapranov.
    Veronese Curves and Grothendieck-Knudsen Moduli Space.
    J. Alg. Geom., 2:239-262, 1993. MR 94a:14024
  • [M] D. Mumford.
    Geometric Invariant Theory.
    Springer-Verlag, 1965. MR 35:5451
  • [Sc] I. Schur.
    Uber die durch collineare grundgebilde erzeugten curven und flachen.
    Math. Ann., 18:1-32, 1881.
  • [SR] J. Semple and S. Roth.
    Introduction to Algebraic Geometry.
    Clarendon Press, 1949. MR 11:535d
  • [Stu] R. Sturm.
    Die Gebilde ersten und zweiten Grades der liniengeometrie in synthetischer Behandlung, Vol. 1.
    B. G. Teubner, 1892.
  • [Sf] B. Sturmfels.
    Algorithms in Invariant Theory.
    Springer-Verlag, 1993. MR 94m:13004
  • [T] J. Todd.
    Configurations defined by six lines.
    Proc. Cambridge Phil. Soc., 29:52-68, 1932.
  • [W] H. Weyl.
    The Classical Groups: Their Invariants and Representations.
    Princeton University Press, 1946. MR 1:42c
  • [Z] O. Zariski.
    Algebraic Surfaces.
    Springer-Verlag, 1995. MR 96c:14024

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14L24, 14Q10

Retrieve articles in all journals with MSC (2000): 14L24, 14Q10

Additional Information

Dana R. Vazzana
Affiliation: Department of Mathematics and Computer Science, Truman State University, Kirksville, Missouri 63501

Keywords: Algebraic geometry, invariant theory
Received by editor(s): July 25, 1999
Published electronically: January 18, 2001
Additional Notes: The author would like to thank Igor Dolgachev for his invaluable assistance in producing this research.
Article copyright: © Copyright 2001 American Mathematical Society