Geometry of bisections of elliptic surfaces and Zariski $N$-plets II

TL;DR

This paper investigates the geometry of bisections on rational elliptic surfaces and constructs specific Zariski N-plets of plane curves, revealing new examples of complex algebraic curve arrangements.

Contribution

It provides new constructions of Zariski N-plets with explicit geometric configurations and extends known results to include a new Zariski 5-plet for degree 8 plane curves.

Findings

Constructed Zariski N+1-plets with specific curve components

Provided explicit examples of complex algebraic curve arrangements

Extended known results to include a new Zariski 5-plet for degree 8

Abstract

In this article, we continue to study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski N + 1-plets of degree 2N + 4 whose irreducible components are an irreducible quartic curve with 2 nodes and N smooth conics. Furthermore, by considering the case of N = 2 and combining with known results, a new Zariski 5-plet for reduced plane curves of degree 8 is given.