The realization space of a certain conic line arrangement of degree 7 and a $\pi_1$-equivalent Zariski pair

TL;DR

This paper investigates the topological diversity of specific degree 7 conic line arrangements, identifying a Zariski pair with identical combinatorics but different topological types, thus revealing multiple connected components in their realization space.

Contribution

It demonstrates the existence of a $ ext{ extpi}_1$-equivalent Zariski pair for certain conic line arrangements of degree 7, expanding understanding of their realization space topology.

Findings

Existence of a $ ext{ extpi}_1$-equivalent Zariski pair.

Determination of the number of connected components in the realization space.

Identification of topologically distinct arrangements with identical combinatorics.

Abstract

In this paper, we continue the study of the embedded topology of plane algebraic curves. We study the realization space of conic line arrangements of degree $7$ with certain fixed combinatorics and determine the number of connected components. This is done by showing the existence of a Zariski pair having these combinatorics, which we identified as a $\pi_1$-equivalent Zariski pair.