Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

TL;DR

This paper introduces a new Zariski pair of degree 8 conic-line arrangements with non-isomorphic fundamental groups and revisits two degree 7 Zariski pairs, providing alternative proofs using fundamental group calculations.

Contribution

The paper constructs a novel Zariski pair of degree 8 arrangements with distinct fundamental groups and offers new proofs for two degree 7 Zariski pairs using the Zariski-van Kampen Theorem.

Findings

New Zariski pair of degree 8 with non-isomorphic fundamental groups

Alternative proofs for two degree 7 Zariski pairs

Use of Coxeter groups in fundamental group determination

Abstract

We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of degree $7$ Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods.