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Elliptic surfaces of rank one and the topology of cubic-line arrangements
Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jnt.2020.06.005
Shinzo Bannai , Hiro-o Tokunaga

Abstract Let φ : S → C be an elliptic surface over a smooth curve C with a section O. We denote its generic fiber by E S which can be considered as an elliptic curve over C ( C ) . For a divisor D on S, not contained in fibers of φ, we canonically associate a C ( C ) -rational point P D of E S . In this note, we give a description of P D , when the rank of the group of C ( C ) -rational points of E S is one. We apply our description to refine our result on a Zariski pair for a cubic-line arrangement.

中文翻译:

一阶椭圆曲面和立方线排列的拓扑结构

摘要 设φ:S→C 是截面为O 的光滑曲线C 上的椭圆曲面。我们用ES 表示它的泛型纤维,可以认为是C(C) 上的椭圆曲线。对于 S 上的除数 D,不包含在 φ 的纤维中,我们规范地关联 ES 的 C ( C ) -有理点 PD。在这篇笔记中,我们描述了 PD ,当 ES 的 C ( C ) -有理点群的秩为 1 时。我们应用我们的描述来改进我们在 Zariski 对上的立方线排列的结果。
更新日期:2021-04-01
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