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A note on elliptic K3 surfaces

Author: JongHae Keum
Journal: Trans. Amer. Math. Soc. 352 (2000), 2077-2086
MSC (2000): Primary 14J28, 14J27, 11H31
Published electronically: November 17, 1999
MathSciNet review: 1707196
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Abstract: We study the relationship between an elliptic fibration on an elliptic K3 surface and its Jacobian surface. We give an explicit description of the Picard lattice of the Jacobian surface. Then we use the description to prove that certain K3 surfaces do not admit a non-Jacobian fibration. Moreover, we obtain an inequality involving the determinant of the Picard lattice and the number of components of reducible fibres, which implies, among others, that if an elliptic K3 surface has Picard lattice with relatively small determinant, then every elliptic fibration on it must have a reducible fibre. Some examples of K3 surfaces are discussed.

References [Enhancements On Off] (What's this?)

  • [1] S. Belcastro, Picard lattices of families of $K3$ surfaces, Ph.D. Thesis, University of Michigan (1997).
  • [2] J. Conway and N. Sloane, Sphere packings, lattices and groups, Springer-Verlag, 1988.MR 89a:11067
  • [3] F. Cossec and I. Dolgachev, Enriques Surfaces I, Birkhäuser, Boston, 1989.MR 90h:14052
  • [4] D. Cox, Mordell-Weil groups of elliptic curves over ${\mathbb{C}}(t)$ with $p_{g}=0$ or $1$, Duke Math. J. 49 (1982), 677-689.MR 84a:14029
  • [5] J. Keum, Two extremal elliptic fibrations on Jacobian Kummer surfaces, manuscripta math. 91 (1996), 369-377; erratum, 94 (1997), 543. MR 97h:14053; MR 98m:14038
  • [6] J. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), 269-288.MR 98e:14039
  • [7] D. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), 105-121.MR 85j:14071
  • [8] S. Mukai, On the moduli space of bundles on $K3$ surfaces, I, Vector Bundles on Algebraic Varieties, Proc. Bombay Conference, 1984, Tata Inst. Fund. Research Studies No.11 (1987), 341-413.MR 88i:14036
  • [9] V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 (1980), 103-167.
  • [10] K. Nishiyama, The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups, Japan. J. Math. 22 (1996), 293-347. MR 97m:14037
  • [11] K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves, J. Math. Soc. Japan 41 (1989), 651-680.MR 90j:14044
  • [12] T. Shioda, On the Mordell-Weil lattices, Com. Math. Univ. St. Pauli 39 (1990), 211-240.MR 91m:14056
  • [13] T. Shioda, Theory of Mordell-Weil lattices, Proc. ICM, Kyoto (1990), 473-489.MR 93k:14046
  • [14] E. Vinberg, The two most algebraic K3 surfaces, Math. Ann. 265 (1983), 1-21.MR 85k:14020

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Additional Information

JongHae Keum
Affiliation: Department of Mathematics, Konkuk University, 93-1 Mojin-dong Kwangjin-gu, Seoul 143-701, Korea

Keywords: Elliptic $K3$ surface, Jacobian surface, Picard lattice, lattice packing
Received by editor(s): October 22, 1997
Published electronically: November 17, 1999
Additional Notes: The research was supported by the Korea Research Foundation (1998) and GARC
Article copyright: © Copyright 2000 American Mathematical Society

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