Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface. (English) Zbl 1139.14032

Kuwata, Masato; Shioda, Tetsuji

Konno, Kazuhiro (ed.) et al., Algebraic geometry in East Asia—Hanoi 2005. Proceedings of the 2nd international conference on algebraic geometry in East Asia, Hanoi, Vietnam, October 10–14, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-45-7/hbk). Advanced Studies in Pure Mathematics 50, 177-215 (2008).

Let \(f: X \to \mathbb P^1\) be an elliptic fibration. Then \(f(x)\) defines a function \(u\) in the function field \(k(X)\). This function \(u\) is called the elliptic parameter. The field \(k(X)\) is also the function field of an elliptic curve \(E/k(u)\), hence \(u\) yields a Weierstrass equation for this fibration. The authors consider the problem given a \(K3\)-surface \(X\), determine all (essentially different) elliptic parameters, i.e., find a Weierstrass equation for each elliptic fibration on \(X\). The author solve this problem for the case that \(X\) is the Kummer surface of \(E\times F\) where \(E\) and \(F\) are non-isogenous elliptic curves and the ground field \(k\) equals \(\mathbb C\). The proof uses the classification of possible elliptic fibrations on \(X\) (up to automorphism) by K. Oguiso [J. Math. Soc. Japan 41, No. 4, 651–680 (1989; Zbl 0703.14024)]. However, the author does not provide explicit equations.
For the entire collection see [Zbl 1135.14003].

Reviewer: Remke Kloosterman (Hannover)

MSC:

14J28	\(K3\) surfaces and Enriques surfaces
14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations

Keywords:

Kummer Surfaces; elliptic fibrations

Citations:

Zbl 0703.14024