Elliptic dihedral covers in dimension 2, geometry of sections of elliptic surfaces, and Zariski pairs for line-conic arrangements

In this article, examples of Zariski pairs $(B_1,B_2)$ satisfying the following condition are given: (i) $\mathrm{deg}{B}_1=\mathrm{deg}{B}_2=7$. (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. In order to construct $B_i$ ($i=1,2$), we make use of geometry of sections of rational elliptic surfaces and their group structure. Dihedral covers play important roles to distinguish the topology of $({\mathbb{P}}^2,B_i)$ $(i = 1, 2)$.