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

2025/07/21 by Cogolludo-Agustín, José I., Elduque, Eva
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15814

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 P²\dashrightarrow \mathbb C\mathbb P¹. 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.

Citations

• Geometric realizability of epimorphisms to curve orbifold groups
• On the topology of fiber-type curves: a Zariski pair of affine nodal curves
• Complements to ample divisors and Singularities
• Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
• Fibred K"ahler and quasi-projective Groups
• Fundamental group of sextics of torus type
• Geometry of cohomology support loci for local systems I

Related

• Homotopy type of complements of fiber-type curves
• On the topology of fiber-type curves: a Zariski pair of affine nodal curves
• A new bound on the number of special fibers in a pencil of curves
• Fundamental Groups of a Class of Rational Cuspidal Plane Curves
• Fibers of pencils of curves on smooth surfaces