Higher Order Degrees of Affine Plane Curve Complements

TL;DR

This paper investigates the properties of higher order degrees in the complements of complex affine plane curves, revealing new restrictions on their fundamental groups and clarifying their relation to Alexander polynomials.

Contribution

It introduces new obstructions for groups realizable as fundamental groups of affine plane curve complements and explores their connection to multivariable Alexander polynomials.

Findings

Finiteness and vanishing properties of higher order degrees are established.

New obstructions on the class of fundamental groups are identified.

Relationship between higher order degrees and Alexander polynomials is clarified.

Abstract

We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as fundamental groups of affine plane curve complements. We also clarify the relationship between the higher order degrees and the multivariable Alexander polynomial of a non-irreducible plane curve.