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

TL;DR

This paper constructs specific Zariski pairs of degree 7 plane curves composed of lines and conics with particular singularities, using elliptic surface geometry and dihedral covers to analyze their topological differences.

Contribution

It introduces new examples of Zariski pairs with degree 7 curves, utilizing elliptic surface sections and dihedral covers to distinguish their topologies.

Findings

Constructed Zariski pairs with degree 7 curves

Used elliptic surface geometry to analyze topology

Applied dihedral covers to differentiate topological types

Abstract

In this article, examples of Zariski pairs $(B_1, B_2)$ satisfying the following condition are given: (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. 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 $({\mathbb P}^2, B_i)$ $(i = 1, 2)$.