The fundamental group of the complement of a generic fiber-type curve

TL;DR

This paper characterizes the fundamental group of the complement of generic fiber-type curves in the complex projective plane, linking it to existing examples and providing a criterion related to Zariski's Problem.

Contribution

It introduces a detailed description of these fundamental groups and offers a new criterion for Zariski's Problem on the commutativity of such groups.

Findings

Fundamental groups of generic fiber-type curve complements are characterized.

Existing examples by Eyral and Oka are connected to these groups.

A criterion for Zariski's Problem on commutativity is established.

Abstract

In this paper we describe and characterize the fundamental group of the complement to generic fiber-type curves, i.e. unions of (the closure of) finitely many generic fibers of a component-free pencil $F=[f:g]:\mathbb C\mathbb P^2\dashrightarrow \mathbb C\mathbb P^1$. We also observe that these groups have already appeared in the literature in examples by Eyral and Oka associated with curves of fiber-type. As a byproduct of our techniques we obtain a criterion to address Zariski's Problem on the commutativity of the fundamental group of the complement of a plane curve.