manuscripta mathematica

, Volume 50, Issue 1, pp 73–132

Projective models of Shioda modular surfaces

  • Wolf Barth
  • Klaus Hulek
Article

Abstract

In this paper we consider divisor classes on elliptic modular surfaces S(n) and their associated linear systems. A principal role is played by divisors I which have the property that nI (resp. n/2I) is linearly equivalent to the sum of the n2 sections if n is odd (resp. even). Our main result is the description of four different projective realizations of S(5). Some results concerning S(3) and S(4) are also discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BHM]
    Barth,W., Hulek,K., Moore,R.: Shioda's modular surface S(5) and the Horrocks-Mumford bundle. To appear in: Proceedings of the Tata conference on algebraic vector bundles over algebraic varieties. Bombay 1984Google Scholar
  2. [BPV]
    Barth,W., Peters,C., Van de Ven,A.: Compact Complex Surfaces. Springer Verlag, Berlin, Heidelberg, New York 1984Google Scholar
  3. [Bu]
    Burns,D.: On the geometry of elliptic modular surfaces and representations of finite groups. In: Springer Lecture Notes in Math., Vol. 1008, 1–29 (1983)Google Scholar
  4. [GR]
    Grauert,H., Remmert,R.: Analytische Stellenalgebren. Grundlehren Math. Wiss. 176, Springer Verlag, Heidelberg 1971Google Scholar
  5. [Hir 1]
    Hirzebruch,F.: Arrangements of lines and algebraic surfaces. In: Progr. in Math. Vol. 36, p.113–140 (1983)Google Scholar
  6. [Hir 2]
    Hirzebruch,F.: Chern numbers of algebraic surfaces. An example. Math. Ann. 266, 351–356 (1984)Google Scholar
  7. [HM]
    Horrocks,G., Mumford,D.: A rank 2 vector bundle on ℙ4 with 15.000 symmetries. Topology 12, 63–81 (1973)Google Scholar
  8. [Hu]
    Hulek,K.: Projective geometry of elliptic curves. Preprint, Providence 1982Google Scholar
  9. [In]
    Inose,H.: On certain Kummer surfaces which can be realized as nonsingular quartics in 1c3. Journal Fac. Sc. Tokyo 23, 545–560 (1976)Google Scholar
  10. [Ish]
    Ishida, M.-N.: Hirzebruch's examples of surfaces of general type with Hirzebruch's examples of surfaces of general type with c12=3c2. In: Springer Lecture Notes in Math., Vol. 1016, 412–431 (1983)Google Scholar
  11. [Mu]
    Mumford,D.: Tata lectures on theta I. Progr.in Math. Vol. 28, Birkhäuser Boston 1983Google Scholar
  12. [Na]
    Naruki,I.: Über die Kleinsche Ikosaeder-Kurve sechsten Grades. Math. Ann. 231, 205–216 (1978)Google Scholar
  13. [Sh 1]
    Shioda,T.: On elliptic modular surfaces. J. Math. Soc. Japan 24, 20–59 (1972)Google Scholar
  14. [Sh 2]
    Shioda,T.: Algebraic cycles on certain K3 surfaces in characteristic p. In: Proc. Int. Congr. on Manifolds, 357–364, Univ. Tokyo Press 1975Google Scholar
  15. [Sh I]
    Shioda,T., Inose,M.: On singular K3 surfaces. In: Complex Analysis & Algebraic Geometry, 119–136, Cambridge Univ. Press 1977Google Scholar
  16. [PS]
    Piatetcky-Shapiro,I., Shafarevich,I.: A Torelli theorem for algebraic surfaces of type K3. English translation: Math. USSR Izv. 5, 547–588 (1971)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Wolf Barth
    • 1
  • Klaus Hulek
    • 1
  1. 1.Mathematisches Institut der Universität Erlangen-NürnbergErlangenBundesrepublik Deutschland

Personalised recommendations