Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

TL;DR

This paper explores the relationship between the fundamental group of quasi-projective varieties and orbifold pencils, revealing new connections involving polynomial solutions and the depth invariant of characters.

Contribution

It extends previous work by linking the fundamental group quotient to orbifold maps and introduces new relations involving polynomial solutions and the depth invariant.

Findings

New relations between fundamental groups and orbifold maps.

Connections between curve equations and polynomial solutions.

Introduction of the depth invariant for character analysis.

Abstract

The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a complement to a plane algebraic curve this provides new relations between the fundamental group, the equation of the curve, and the existence of polynomial solutions to certain equations generalizing Pell's equation. These relations are formulated in terms of the depth which is an invariant of the characters of the fundamental group discussed in detail here.