Depth of characters of curve complements and orbifold pencils

TL;DR

This paper explores the use of orbifold pencils to analyze the character varieties of plane curve complements and hyperplane arrangements, providing new geometric tools to understand their properties and distinguish complex examples.

Contribution

It introduces a geometric approach using orbifold pencils to study characters of plane curve complements, replacing traditional fundamental group computations for certain cases.

Findings

Constructed an infinite family of curves with characters of any torsion and depth 3.

Demonstrated how orbifold pencils can replace group-theoretic methods in studying finite order characters.

Revisited a Zariski pair and showed how orbifold pencils distinguish the curves.

Abstract

The present work is a user's guide to the results of a previous paper by the second and third authors, where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils. Below we consider the case of plane curve complements and hyperplane arrangements. In particular, an infinite family of curves exhibiting characters of any torsion and depth~3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.