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arXiv · 2507.15814

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

José I. Cogolludo-Agustín·Eva Elduque
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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.

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José I. Cogolludo-Agustín, Eva Elduque. 2025-07-21. The fundamental group of the complement of a generic fiber-type curve. https://arxiv.org/abs/2507.15814

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