Albanese varieties of cyclic covers of the projective plane and orbifold pencils

TL;DR

This paper explores the relationship between the fundamental group of plane singular curve complements and orbifold pencils, utilizing Albanese varieties of cyclic covers to establish conditions for curve inclusion in orbifold pencils.

Contribution

It introduces new criteria linking fundamental groups and Albanese varieties to orbifold pencils, with explicit examples illustrating the necessity of these conditions.

Findings

Established sufficient conditions for curves to belong to orbifold pencils.

Constructed an example of a cyclic cover as an abelian surface related to a genus 2 curve.

Demonstrated the role of Albanese varieties in understanding the topology of plane curve complements.

Abstract

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.