Detecting Zariski Pairs by Algorithms and Computational Classification in Conic Line Arrangements
Meirav Amram, Gal Goren
TL;DR
The paper addresses detecting Zariski pairs in conic–line arrangements by reframing the tubular-neighborhood homeomorphism condition into a combinatorial criterion, enabling systematic classification into combinatorial types. It develops an inductive framework to generate all $(n,1)$-arrangements up to combinatorial equivalence, and analyzes each type for potential Zariski pairs using structural lemmas, projective equivalence via $PGL(3,\mathbb{C})$, and fundamental group calculations of curve-complement groups through the Zariski–van Kampen theorem. A key result is that no $(n,1)$-arrangements form a Zariski pair for $n\le 4$, with a complete classification achieved up to $n\le 5$, and ongoing plans to extend to larger $n$ and to configurations with multiple conics. The framework provides a computational path from combinatorial data to topological distinguishability, leveraging group-theoretic invariants to certify Zariski pairs and guide the search for new examples in conic–line geometry.
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.