Abstract
A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the 16 curves. In the paper [16], we constructed explicitly non-isomorphic Kummer structures on some Kummer surfaces. In this paper we generalize the construction to Kummer surfaces with a weaker restriction on the degree of the polarization and we describe some cases where the previous construction does not work.
Acknowledgements
We thank P. Stellari for pointing out his paper [20]. We also thank K. Hulek, H. Lange, K. Oguiso, M. Ramponi, J. Rivat and T. Shioda for useful discussions. We are very grateful to the referee for the many questions, remarks and comments that improved this article substantially.
Communicated by: M. Joswig
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Appendix
Why it was natural to study the case t =
Thus 2t ⋅ 4k(k + 1) is the square of an integer and it is therefore natural to define t =
A table. We enumerate in the following table the fundamental solutions (α0,β0) of the Pell–Fermat equations α2 − 2tβ2 = 1 for 2t ≤ 60. Recall that there are non-trivial solutions if and only if 2t is not a square.
When 2t = k(k + 1) the minimal solution is (2k + 1, 2); these correspond to Nikulin configurations studied in [16], and we put a * close to these cases. We put a box around the cases with β0 odd, and a prime ′ when β0 is even but such that the negative Pell–Fermat equation has a solution: these cases are left out in this paper.
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