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Transactions of the American Mathematical Society

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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
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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] Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
  • [D] Igor V. Dolgachev, Introduction to geometric invariant theory, Lecture Notes Series, vol. 25, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. MR 1312159
  • [GH] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • [Gr] F. Grosshans, R. Gleeson, M. Hirsch, R. M. Williams.
    Object-image equations for x points and 6 - x lines.
    Preprint, 1998.
  • [Ha] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [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] Bruce Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Springer-Verlag, Berlin, 1996. MR 1438547
  • [Ht2]
  • [K] M. M. Kapranov, Veronese curves and Grothendieck-Knudsen moduli space \overline𝑀_{0,𝑛}, J. Algebraic Geom. 2 (1993), no. 2, 239–262. MR 1203685
  • [M] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602
  • [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] Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980
  • [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] Oscar Zariski, Algebraic surfaces, Classics in Mathematics, Springer-Verlag, Berlin, 1995. With appendices by S. S. Abhyankar, J. Lipman and D. Mumford; Preface to the appendices by Mumford; Reprint of the second (1971) edition. MR 1336146

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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