On fundamental groups of plane curve complements

TL;DR

This paper explores the properties of fundamental groups and Alexander polynomials of plane curves, examining their relationships and topological characteristics, including a modern reinterpretation of Zariski's results.

Contribution

It provides new insights into the connection between Alexander polynomials and (nearly) free irreducible plane curves, and revisits classical results with modern methods.

Findings

Non-trivial Alexander polynomials relate to (nearly) freeness of curves

Reinterpretation of Zariski's classical results

Topological properties of curves with abelian fundamental groups

Abstract

In this paper we discuss some properties of fundamental groups and Alexander polynomials of plane curves. We discuss the relationship of the non-triviality of Alexander polynomials and the notion of (nearly) freeness for irreducible plane curves. We reprove and restate in modern terms a somewhat forgotten result of Zariski. Finally, we describe some topological properties of curves with abelian fundamental group.