Cubic surfaces with a Galois invariant double-six

Andreas-Stephan Elsenhans; Jörg Jahnel

Open Mathematics (2010)

  • Volume: 8, Issue: 4, page 646-661
  • ISSN: 2391-5455

Abstract

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We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.

How to cite

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Andreas-Stephan Elsenhans, and Jörg Jahnel. "Cubic surfaces with a Galois invariant double-six." Open Mathematics 8.4 (2010): 646-661. <http://eudml.org/doc/269074>.

@article{Andreas2010,
abstract = {We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.},
author = {Andreas-Stephan Elsenhans, Jörg Jahnel},
journal = {Open Mathematics},
keywords = {Cubic surface; Hexahedral form; Double-six; Explicit Galois descent; cubic surface; double-six; hexahedral form; Galois descent},
language = {eng},
number = {4},
pages = {646-661},
title = {Cubic surfaces with a Galois invariant double-six},
url = {http://eudml.org/doc/269074},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Andreas-Stephan Elsenhans
AU - Jörg Jahnel
TI - Cubic surfaces with a Galois invariant double-six
JO - Open Mathematics
PY - 2010
VL - 8
IS - 4
SP - 646
EP - 661
AB - We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.
LA - eng
KW - Cubic surface; Hexahedral form; Double-six; Explicit Galois descent; cubic surface; double-six; hexahedral form; Galois descent
UR - http://eudml.org/doc/269074
ER -

References

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  1. [1] Bourbaki N., Éléments de Mathématique, Livre II: Algèbre, Chapitre V, Masson, Paris, 1981 
  2. [2] Coble A.B., Point sets and allied Cremona groups I, Trans. Amer. Math. Soc., 1915, 16(2), 155–198 Zbl45.0940.13
  3. [3] Cremona L., Ueber die Polar-Hexaeder bei den Flächen dritter Ordnung, Math. Ann., 1878, 13(2), 301–304 http://dx.doi.org/10.1007/BF01446536[Crossref] Zbl10.0418.02
  4. [4] Dolgachev I.V., Topics in classical algebraic geometry. Part I, preprint available at http://www.math.lsa.umich.edu/˜idolga/topics1.pdf Zbl0284.14016
  5. [5] Ekedahl T., An effective version of Hilbert’s irreducibility theorem, In: Séminaire de Théorie des Nombres (1988–1989 Paris), Progr. Math., 91, Birkhäuser, Boston, 1990, 241–249 
  6. [6] Elsenhans A.-S., Good models for cubic surfaces, preprint available at http://www.staff.uni-bayreuth.de/˜btm216/red_5.pdf 
  7. [7] Elsenhans A.-S., Jahnel J., Experiments with general cubic surfaces, In: Algebra, Arithmetic and Geometry - Manin Festschrift, (in press), preprint available at http://www.math.nyu.edu/˜tschinke/.manin/submitted/jahnel.pdf 
  8. [8] Elsenhans A.-S., Jahnel J., The discriminant of a cubic surface, preprint available at http://www.uni-math.gwdg.de/jahnel/Preprints/Oktik7.ps 
  9. [9] Glauberman G., On the Suzuki groups and the outer automorphisms of S 6, In: Groups, difference sets, and the Monster (1993 Columbus/Ohio), de Gruyter, Berlin, 1996, 55–72 Zbl0846.20023
  10. [10] Hartshorne R., Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, New York-Heidelberg, 1977 
  11. [11] Malle G., Matzat B.H., Inverse Galois Theory, Springer, Berlin, 1999 Zbl0940.12001
  12. [12] Manin Y.I., Cubic Forms: Algebra, Geometry, Arithmetic, Elsevier, New York, 1974 
  13. [13] Reye T., Über Polfünfecke und Polsechsecke räumlicher Polarsysteme, J. Reine Angew. Math., 1874, 77, 269–288 
  14. [14] Schläfli L., An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface, Quart. J. Math., 1858, 2, 110–120 
  15. [15] Schläfli L., Quand’è che dalla superficie generale di terzo ordine si stacca una parte che non sia realmente segata da ogni piano reale?, Ann. Math. Pura Appl., 1872, 5, 289–295 [Crossref] Zbl05.0321.01
  16. [16] Serre J.-P., Groupes Algébriques et Corps de Classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 Zbl0097.35604

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