Irreducible plane sextics with large fundamental groups

TL;DR

This paper computes the fundamental groups and Alexander modules of certain irreducible plane sextics with large weights, providing detailed geometric descriptions and exploring their moduli spaces.

Contribution

It presents the first comprehensive calculation of fundamental groups and Alexander modules for irreducible sextics of weights eight and nine, including derivatives.

Findings

Fundamental groups of these sextics are computed.

Alexander modules are shown to be free over specific rings.

Detailed geometric and moduli space descriptions are provided.

Abstract

We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.