A Structure Theorem for Fibrations on Delsarte Surfaces

Part of the Fields Institute Communications book series (FIC, volume 67)


In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change, the fibration is the pullback of a fibration with three singular fibers and that this second-base change is completely ramified at two points where the fiber is singular. As a corollary we show that every Delsarte fibration of genus 1 with nonconstant j-invariant occurs as the base change of an elliptic surface from Fastenberg’s list of rational elliptic surfaces with γ < 1.

Key words

Delsarte surfaces Elliptic surfaces 

Mathematics Subject Classifications (2010)

Primary 14J27 Secondary 14J25 



This paper is inspired by the results of [4, Chap.  6] and by some discussions which took place on the occasion of the Ph.D. defense of the first author at the University of Groningen. Part of the research was done while the first author held a position at the University of Groningen. His position was supported by a grant of the Netherlands Organization for Scientific Research (NWO). The research is partly supported by ERC Starting grant 279723 (SURFARI). The second author acknowledges the hospitality of the University of Groningen and the Leibniz Universität Hannover, where most of the work was done.

We thank the referee for providing many comments to improve the exposition.


  1. 1.
    J. Chahal, M. Meijer, J. Top, Sections on certain j = 0 elliptic surfaces. Comment. Math. Univ. St. Paul. 49(1), 79–89 (2000)MathSciNetMATHGoogle Scholar
  2. 2.
    L.A. Fastenberg, Computing Mordell–Weil ranks of cyclic covers of elliptic surfaces. Proc. Am. Math. Soc. 129(7), 1877–1883 (electronic) (2001)Google Scholar
  3. 3.
    L.A. Fastenberg, Cyclic covers of rational elliptic surfaces. Rocky Mt. J. Math. 39(6), 1895–1903 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    B. Heijne, Elliptic Delsarte surfaces, Ph.D. thesis, Rijksuniversiteit Groningen, 2011Google Scholar
  5. 5.
    B. Heijne, The maximal rank of elliptic Delsarte surfaces. Math. Comp. 81(278), 1111–1130 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R. Kloosterman, O. Tommasi, Locally trivial families of hyperelliptic curves: the geometry of the Weierstrass scheme. Indag. Math. (N.S.) 16(2), 215–223 (2005)Google Scholar
  7. 7.
    T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces. Am. J. Math. 108(2), 415–432 (1986)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    T. Shioda, Some remarks on elliptic curves over function fields. Astérisque, 209(12), 99–114, 1992. Journées Arithmétiques, 1991 (Geneva)Google Scholar
  9. 9.
    J. Steenbrink, Intersection form for quasi-homogeneous singularities. Compos. Math. 34(2), 211–223 (1977)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut für algebraische GeometrieLeibniz Universität HannoverHannoverGermany
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations