Plane sextics with a type $\bold E_8$ singular point

TL;DR

This paper constructs explicit models and computes fundamental groups of plane sextic curves with simple singularities, focusing on those with an _8 singularity, revealing new nonabelian fundamental groups and finite groups.

Contribution

It provides explicit geometric models and fundamental group computations for plane sextics with _8 singularities, including four newly discovered sextics with nonabelian groups.

Findings

Discovered four new sextics with nonabelian fundamental groups

Two irreducible sextics have finite fundamental groups

Utilized reduction to trigonal curves and dessins d'enfants

Abstract

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's {\it dessins d'enfants}