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

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.

Includes an appendix with the full computation of the fundamental groups and a similar computations for Zariski pairs of conic-line arrangements discovered by the last named author in 2014

References in corpus (4)

• Conic-line arrangements in the complex projective plane
• Torsion divisors of plane curves and Zariski pairs
• Torsion divisors of plane curves with maximal flexes and Zariski pairs
• An explicit construction for $n$-contact curves to a smooth cubic via divisions and Zariski tuples

Cited by in corpus (1)

• Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture