The fundamental group of the complement of a generic fiber‐type curve
José I. Cogolludo‐Agustín, Eva Elduque
Source abstract
Abstract In this paper, we describe and characterize the fundamental group of the complement of generic fiber‐type curves, that is, unions of (the closure of) finitely many generic fibers of a component‐free pencil . 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 prove that the fundamental group of a generic fiber‐type curve depends only on the number of irreducible components, their degree and the degree and number of multiple fibers in its pencil. We also obtain new criteria to address Zariski's Problem on the commutativity of the fundamental group of the complement of a plane curve.
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