Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers

Abstract

In this article, we make use of geometry of sections of elliptic surfaces and elementary arithmetic on the Mordell-Weil group in order to study existence problem of dihedral covers with given reduced curves as the branch loci. As an application, we give some examples of Zariski pairs (B₁, B₂) for “conic-line arrangements” satisfying the following conditions:
(i) deg B₁ = deg B₂ = 7.
(ii) Irreducible components of B_i (i = 1, 2) are lines and conics.
(iii) Singularities of B_i (i = 1, 2) are nodes, tacnodes and ordinary triple points.