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 $(B_1, B_2)$ for “conic-line arrangements” satisfying the following conditions:

(i) $\deg B_1 = \deg B_2 = 7$.

(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.