Zariski $N$-ples for a smooth cubic and its tangent lines

TL;DR

This paper investigates the geometry of elliptic curve two-torsion points to differentiate the embedded topology of reducible plane curves made of a smooth cubic and tangent lines, introducing a new family of Zariski N-ples.

Contribution

It presents a novel approach using elliptic curve torsion points to classify the topology of specific reducible plane curves, leading to new Zariski N-ples.

Findings

New family of Zariski N-ples identified

Method distinguishes embedded topologies of cubic-tangent line arrangements

Elliptic curve torsion points are key to classification

Abstract

In this paper, we study the geometry of two-torsion points of elliptic curves in order to distinguish the embedded topology of reducible plane curves consisting of a smooth cubic and its tangent lines. As a result, we obtain a new family of Zariski N-ples consisting of such curves.