Detecting Zariski Pairs by Algorithms and Computational Classification in Conic Line Arrangements

TL;DR

This paper introduces an algorithmic approach to identify Zariski pairs in conic line arrangements by classifying arrangements combinatorially and verifying potential pairs through topological and algebraic invariants.

Contribution

It develops a new combinatorial condition and an inductive algorithm for classifying conic line arrangements, facilitating the detection of Zariski pairs.

Findings

Successfully classifies arrangements into equivalence classes

Identifies potential Zariski pairs using structural lemmas

Computes fundamental groups via Zariski van Kampen Theorem

Abstract

We present an approach to detecting Zariski pairs in conic line arrangements. Our method introduces a combinatorial condition that reformulates the tubular neighborhood homeomorphism criterion arising in the definition of Zariski pairs. This allows for a classification of arrangements into combinatorial equivalence classes, which we generate systematically via an inductive algorithm. For each class, potential Zariski pairs are examined using structural lemmas, projective equivalence, and fundamental group computations obtained through the Zariski van Kampen Theorem.