On the connection between fundamental groups and pencils with multiple fibers

TL;DR

This paper explores the relationship between fundamental groups of quasiprojective manifolds and linear systems, presenting new examples that challenge existing assumptions about mappings and characteristic varieties.

Contribution

It provides two novel results: a plane curve with a non-abelian fundamental group that cannot be mapped onto a non-abelian orbifold, and an affine manifold with characteristic varieties not derived from orbifold pull-backs.

Findings

Existence of a plane curve with non-abelian fundamental group and no orbifold mapping

An affine manifold with characteristic varieties not from orbifold pull-backs

Challenges previous beliefs about fundamental groups and linear systems

Abstract

We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold.