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.
@article{bwmeta1.element.doi-10_2478_s11533-010-0036-1,
author = {Andreas-Stephan Elsenhans and J\"org Jahnel},
title = {Cubic surfaces with a Galois invariant double-six},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {646-661},
zbl = {1203.14040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0036-1}
}
Andreas-Stephan Elsenhans; Jörg Jahnel. Cubic surfaces with a Galois invariant double-six. Open Mathematics, Tome 8 (2010) pp. 646-661. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0036-1/
[1] Bourbaki N., Éléments de Mathématique, Livre II: Algèbre, Chapitre V, Masson, Paris, 1981
[2] Coble A.B., Point sets and allied Cremona groups I, Trans. Amer. Math. Soc., 1915, 16(2), 155–198 | Zbl 45.0940.13
[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] | Zbl 10.0418.02
[4] Dolgachev I.V., Topics in classical algebraic geometry. Part I, preprint available at http://www.math.lsa.umich.edu/˜idolga/topics1.pdf | Zbl 0284.14016
[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] Elsenhans A.-S., Good models for cubic surfaces, preprint available at http://www.staff.uni-bayreuth.de/˜btm216/red_5.pdf
[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] 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] 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 | Zbl 0846.20023
[10] Hartshorne R., Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, New York-Heidelberg, 1977
[11] Malle G., Matzat B.H., Inverse Galois Theory, Springer, Berlin, 1999 | Zbl 0940.12001
[12] Manin Y.I., Cubic Forms: Algebra, Geometry, Arithmetic, Elsevier, New York, 1974
[13] Reye T., Über Polfünfecke und Polsechsecke räumlicher Polarsysteme, J. Reine Angew. Math., 1874, 77, 269–288
[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] 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] | Zbl 05.0321.01
[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 | Zbl 0097.35604