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

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.