Elliptic surfaces of rank one and the topology of cubic-line arrangements

S.Bannai and H.Tokunaga

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.

Journal of Number Theory

221

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