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

2021/06/07 by Meirav Amram, Robert Shwartz, Amram, Meirav +6 · 2 citations
Computer Science · Mathematics · #14H30 (Primary) 14H50 #14Q05 #20F55 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2106.03507

openalex publication_date 2021/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• 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

• Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8
• On plus-one generated arrangements of plane conics

Related

• The realization spaces of certain conic-line arrangements of degree 7
• Zariski pairs of conic-line arrangements with a unique conic
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• The realization space of a certain conic line arrangement of degree 7 and a π₁-equivalent Zariski pair
• Fundamental groups of Tangent Conic-Line arrangements with Singularities up to order 6