Advertisement

Geometriae Dedicata

, Volume 159, Issue 1, pp 29–40 | Cite as

The discriminant of a cubic surface

  • Andreas-Stephan Elsenhans
  • Jörg Jahnel
Original Paper

Abstract

The 27 lines on a smooth cubic surface over \({{\mathbb Q}}\) are acted upon by a finite quotient of \({{\rm Gal}(\overline{\mathbb Q}/{\mathbb Q})}\) . We construct explicit examples such that the operation is via the index two subgroup of the maximal possible group. This is the simple group of order 25,920. Our examples are given in pentahedral normal form with rational coefficients. For such cubic surfaces, we study the discriminant and show its relation to the index two subgroup.

Keywords

Cubic surface Pentahedral normal form Discriminant Rational point 

Mathematics Subject Classification (2000)

14J45 14J20 11G35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexander J., Hirschowitz A.: Polynomial interpolation in several variables. J. Algebraic Geom. 4, 201–222 (1995)MathSciNetMATHGoogle Scholar
  2. 2.
    Burkhardt H.: Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen III. Math. Ann. 41, 313–343 (1893)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Clebsch A.: Ueber eine Transformation der homogenen Functionen dritter Ordnung mit vier Veränderlichen. J. für die reine und angew. Math. 58, 109–126 (1861)MATHCrossRefGoogle Scholar
  4. 4.
    Conrad B.: Grothendieck Duality and Base Change. Lecture Notes in Mathematics, vol. 1750. Springer, Berlin (2000)CrossRefGoogle Scholar
  5. 5.
    Dardanelli E., van Geemen B.: Hessians and the moduli space of cubic surfaces. In: Keum, J., Kondo, S (eds) Algebraic Geometry, Contemporary Mathematics, vol 422., pp. 17–36. AMS, Providence (2007)Google Scholar
  6. 6.
    Edge W.L.: The discriminant of a cubic surface. Proc. Roy. Irish Acad. 80, 75–78 (1980)MathSciNetMATHGoogle Scholar
  7. 7.
    Elsenhans A.-S., Jahnel J.: Experiments with general cubic surfaces. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry, In Honor of Yu., I. Manin, Volume I, Progress in Mathematics, vol. 269, pp. 637–654. Birkhäuser, Boston (2007)Google Scholar
  8. 8.
    Elsenhans, A.-S., Jahnel, J.: On the arithmetic of the discriminant for cubic surfaces. http://www.uni-math.gwdg.de/jahnel/Preprints/OktikB_1b.pdf
  9. 9.
    Grothendieck A.: Revêtements étales et groupe fondamental (SGA 1). Lecture Notes Mathematics, vol. 224. Springer, Berlin (1971)Google Scholar
  10. 10.
    Jordan C.: Sur la trisection des fonctions abéliennes et sur les 27 droites des surfaces du troisième ordre. C. R. Acad. Sci. Paris 68, 865–869 (1869)Google Scholar
  11. 11.
    Matsumura H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1989)Google Scholar
  12. 12.
    Mella M.: Singularities of linear systems and the Waring problem. Trans. Am. Math. Soc. 358, 5523–5538 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ranestand K., Scgreyer F.-O.: Varieties of sums of powers. J. für Die Reine Und Angew. Math. 525, 147–181 (2000)CrossRefGoogle Scholar
  14. 14.
    Salmon G.: A treatise on the analytic geometry of three dimensions. Hodges/Smith, Dublin (1865)Google Scholar
  15. 15.
    Sylvester J.J.: Sketch of a memoir on elimination, transformation, and canonical forms. Camb. Dublin Math. J. 6, 186–200 (1851)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Département MathematikUniversität SiegenSiegenGermany

Personalised recommendations