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.

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Additional Information

**JongHae Keum**

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

Email:
jhkeum@kkucc.konkuk.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-99-02587-8

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