Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2012 | 10 | 3 | 903-926
Tytuł artykułu

On the order three Brauer classes for cubic surfaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.
Opis fizyczny
  • [1] Buckley A., Košir T., Determinantal representations of smooth cubic surfaces, Geom. Dedicata, 2007, 125, 115–140
  • [2] Cartan H., Eilenberg S., Homological Algebra, Princeton University Press, Princeton, 1956
  • [3] Cassels J.W.S., Guy M.J.T., On the Hasse principle for cubic surfaces, Mathematika, 1966, 13, 111–120
  • [4] Clebsch A., Die Geometrie auf den Flächen dritter Ordnung, J. Reine Angew. Math., 1866, 1866(65), 359–380
  • [5] Colliot-Thélène J.-L., Kanevsky D., Sansuc J.-J., Arithmétique des surfaces cubiques diagonales, In: Diophantine Approximation and Transcendence Theory, Bonn, May–June, 1985, Lecture Notes in Math., 1290, Springer, Berlin, 1987, 1–108
  • [6] Cremona L., Sulle ventisette rette della superficie del terzo ordine, Istit. Lombardo Accad. Sci. Lett. Rend. A, 1870, 3, 209–219
  • [7] Dolgachev I.V., Classical Algebraic Geometry, Cambridge University Press (in press), available at
  • [8] Elsenhans A.-S., Jahnel J., Experiments with general cubic surfaces, In: Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, 1, Progr. Math., 269, Birkhäuser, Boston, 2007, 637–653
  • [9] Elsenhans A.-S., Jahnel J., On the Brauer-Manin obstruction for cubic surfaces, J. Comb. Number Theory, 2010, 2(2), 107–128
  • [10] Elsenhans A.-S., Jahnel J., Cubic surfaces with a Galois invariant pair of Steiner trihedra, Int. J. Number Theory, 2011, 7(4), 947–970
  • [11] Elsenhans A.-S., Jahnel J., On the quasi group of a cubic surface over a finite field, J. Number Theory (in press)
  • [12] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Heidelberg, 1977
  • [13] Henderson A., The Twenty-Seven Lines upon the Cubic Surface, Hafner, New York, 1960
  • [14] Jahnel J., More cubic surfaces violating the Hasse principle, J. Théor. Nombres Bordeaux, 2011, 23(2), 471–477
  • [15] Kresch A., Tschinkel Yu., On the arithmetic of del Pezzo surfaces of degree 2, Proc. London Math. Soc., 2004, 89(3), 545–569
  • [16] Kunyavskiĭ B.È., Skorobogatov A.N., Tsfasman M.A., Del Pezzo Surfaces of Degree Four, Mém. Soc. Math. France (N.S.), 37, Société Mathématique de France, Marseille, 1989
  • [17] Manin Yu.I., Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Library, 4, North-Holland/Elsevier, Amsterdam-London/New York, 1974
  • [18] Mordell L.J., On the conjecture for the rational points on a cubic surface, J. London Math. Soc., 1965, 40, 149–158
  • [19] Neukirch J., Klassenkörpertheorie, 2nd ed., Springer-Lehrbuch, Springer, Berlin, 2011
  • [20] Peyre E., Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J., 1995, 79(1), 101–218
  • [21] Peyre E., Tschinkel Yu., Tamagawa numbers of diagonal cubic surfaces, numerical evidence, Math. Comp., 2001, 70(233), 367–387
  • [22] 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, The Quarterly Journal of Pure and Applied Mathematics, 1858, 2, 110–120
  • [23] Serre J.-P., Local class field theory, In: Algebraic Number Theory, Brighton, 1965, Thompson, Washington, 1967, 128–161
  • [24] Steiner J., Über die Flächen dritten Grades, J. Reine Angew. Math., 1857, 53, 133–141
  • [25] Swinnerton-Dyer H.P.F., Two special cubic surfaces, Mathematika, 1962, 9, 54–56
  • [26] Swinnerton-Dyer H.P.F., Universal equivalence for cubic surfaces over finite and local fields, In: Symposia Mathematica, 24, Rome, April 9–14, 1979, Academic Press, London-New York, 1981, 111–143
  • [27] Swinnerton-Dyer P., The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc., 1993, 113(3), 449–460
  • [28] Tate J.T., Global class field theory, In: Algebraic Number Theory, Brighton, 1965, Thompson, Washington, 1967, 162–203
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.