Archiv der Mathematik

, Volume 98, Issue 3, pp 229–234 | Cite as

On cubic surfaces with a rational line

Article

Abstract

We report on our project to construct non-singular cubic surfaces over \({\mathbb{Q}}\) with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.

Mathematics Subject Classification (2010)

Primary 14J26 Secondary 14G25 11G35 

Keywords

Cubic surface Degree 4 Del Pezzo surface Explicit Galois descent 

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References

  1. 1.
    A.-S. Elsenhans, Good models for cubic surfaces, Preprint.Google Scholar
  2. 2.
    Elsenhans A.-S., Jahnel J.: Cubic surfaces with a Galois invariant double-six. Central European Journal of Mathematics 8, 646–661 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A.-S. Elsenhans and J. Jahnel, Cubic surfaces with a Galois invariant pair of Steiner trihedra, to appear in: International Journal of Number Theory.Google Scholar
  4. 4.
    Kunyavskij B. È., Skorobogatov A. N., Tsfasman M. A.: Del Pezzo surfaces of degree four. Mém. Soc. Math. France 37, 1–113 (1989)MATHGoogle Scholar
  5. 5.
    MalleG. Matzat B.H.: Inverse Galois theory. Springer, Berlin (1999)Google Scholar
  6. 6.
    Yu. I. Manin, Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974.Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

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

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