Mathematische Annalen

, Volume 328, Issue 3, pp 451–468

Supersingular K3 surfaces in odd characteristic and sextic double planes

Article

Abstract

We show that every supersingular K3 surface is birational to a double cover of a projective plane.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, M.: Supersingular K3 surfaces. Ann. Sci. École Norm. Sup. (4) 7(1974), 543–567 (1975)Google Scholar
  2. 2.
    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV-Chapitre VI. Hermann, Paris, 1968Google Scholar
  3. 3.
    Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups. Third ed., Springer-Verlag, New York, 1999Google Scholar
  4. 4.
    Ebeling, W.: Lattices and codes. Friedr. Vieweg & SohnGoogle Scholar
  5. 5.
    Ibukiyama, T.: A basis for the algebra of quaternions over the field of rational numbers and its maximal orders. in Japanese, Sûgaku 24, 316–318 (1972)Google Scholar
  6. 6.
    Nikulin, V.V.: Weil linear systems on singular K3 surfaces. Algebraic geometry and analytic geometry (Tokyo, 1990), Springer, Tokyo, 1991, pp. 138–164Google Scholar
  7. 7.
    Rudakov, A.N., Šafarevič, I.R.: Supersingular K3 surfaces over fields of characteristic 2 Izv. Akad. Nauk SSSR Ser. Mat. 42(4), 848–869 (1978); Igor~R.Shafarevich, Collected mathematical papers, Springer-Verlag, Berlin, 1989, pp. 614–632MATHGoogle Scholar
  8. 8.
    Rudakov, A.N., Šafarevič, I.R.: Surfaces of type K3 over fields of finite characteristic. Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 115–207; Igor~R.Shafarevich, Collected mathematical papers, Springer-Verlag, Berlin, 1989, pp. 657–714Google Scholar
  9. 9.
    Serre, J.-P.: A course in arithmetic. Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York, 1973Google Scholar
  10. 10.
    Shimada, I.: Rational double points on supersingular K3 surfaces. Preprint 2003, math.AG/0311057. To appear in Math. Comp.Google Scholar
  11. 11.
    Shioda, T.: Supersingular K3 surfaces. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., Vol. 732, Springer, Berlin, 1979, pp. 564–591Google Scholar
  12. 12.
    Urabe, T.: Dynkin graphs and combinations of singularities on plane sextic curves. Singularities (Iowa City, IA, 1986), Amer. Math. Soc. Providence, RI, 1989, pp. 295–316Google Scholar
  13. 13.
    Venkov, B.B.: On the classification of integral even unimodular 24-dimensional quadratic forms. Trudy Mat. Inst. Steklov. 148, 65–76 (1978)MathSciNetMATHGoogle Scholar
  14. 14.
    Yang, J.-G.: Sextic curves with simple singularities. Tohoku Math. J. 48 2(2), 203–227 (1996)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Division of Mathematics, Graduate School of ScienceHokkaido UniversitySapporoJapan

Personalised recommendations