manuscripta mathematica

, Volume 125, Issue 3, pp 325–343 | Cite as

Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, II

Article

Abstract

We construct examples of supersingular Calabi-Yau threefolds in characteristic 2 making use of the method by Schoen. Unirational Calabi-Yau threefolds of five different topological types are obtained. There are two examples with the third Betti number zero among them, and they are counted as other examples of non-liftable Calabi-Yau threefolds in characteristic 2 after the one given by Schröer.

Mathematics Subject Classification (2000)

14J30 14J32 14B05 14D06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin M. (1977). Coverings of the rational double points in characteristic p. In: Baily W.L. Jr., Shioda, T. (eds.) Complex Analysis and Algebraic Geometry, pp 11–22. Iwanami Shoten, Tokyo, Cambridge University Press, Cambridge Google Scholar
  2. 2.
    Artin M. and Mazur B. (1977). Formal groups arising from algebraic varieties. Ann. Sci. École Norm. Sup. 10(4): 87–131 MATHMathSciNetGoogle Scholar
  3. 3.
    Bombieri E. and Mumford D. (1976). Enriques’ classification of surfaces in char. p, III. Invent. Math. 35: 197–232 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bombieri E. and Mumford D. (1977). Enriques’ classification of surfaces in char. p, II. In: Baily,W.L. Jr., Shioda, T. (eds.) Complex Analysis and Algebraic Geometry, pp. 23–42. Iwanami Shoten, Tokyo, Princeton University Press, Princeton Google Scholar
  5. 5.
    Hirokado M. (1999). A non-liftable Calabi-Yau threefold in characteristic 3. Tohoku Math. J. 51(2): 479–487 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hirokado M. (2001). Calabi-Yau threefolds obtained as fiber products of elliptic and quasi-elliptic rational surfaces. J. Pure Appl. Algebra 162: 251–271 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hirokado M. (2004). Deformations of rational double points and simple elliptic singularities in characteristic p. Osaka J. Math. 41: 605–616 MATHMathSciNetGoogle Scholar
  8. 8.
    Hirokado M., Ito H. and Saito N. (2007). Calabi-Yau threefolds arising from the fiber products of rational quasi-elliptic surfaces, I. Ark. Mat. 45(2): 279–296 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Ido, Y.: Construction of unirational Calabi-Yau threefolds as fiber products of rational quasi-elliptic surfaces. Master’s thesis, University of Tokyo (2006)Google Scholar
  10. 10.
    Ito H. (1994). The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic 2. Tôhoku Math. J. 46: 221–251 MATHCrossRefGoogle Scholar
  11. 11.
    Kollár J. and Mori S. (1998). Birational Geometry of Algebraic Geometry. Cambridge University Press, Cambridge MATHGoogle Scholar
  12. 12.
    Nygaard N. (1978). On the fundamental group of a unirational 3-fold. Invent. Math. 44: 75–86 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Reid, M.: Canonical 3-folds. In: Beauville, A. (ed.) Journées de géométrie algébrique d’Angers, pp. 273–310. Sijthoff and Nordhoff (1980)Google Scholar
  14. 14.
    Rudakov A.N. and Shafarevich I.R. (1978). Supersingular K3 surfaces over fields of characteristic 2. Izv. Akad. Nauk SSSR Ser. Mat. 42: 848–869 MATHMathSciNetGoogle Scholar
  15. 15.
    Schoen C. (1988). On fiber products of rational elliptic surfaces with section. Math. Z. 197: 177–199 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schröer S. (2004). Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors. Compositio Math. 140: 1579–1592 MATHGoogle Scholar
  17. 17.
    Schröer, S.: Singularities appearing on generic fibers of morphisms between smooth schemes. Michigan Math. J. (to appear)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Graduate School of Information SciencesHiroshima City UniversityAsaminami-ku, HiroshimaJapan
  2. 2.Department of Applied Mathematics, Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

Personalised recommendations