Plane sextics via dessins d'enfants

TL;DR

This paper introduces a geometric method to analyze plane sextic curves with a triple singularity, providing explicit descriptions and fundamental groups for maximal sextics with an E7 singularity, revealing all groups are finite including a nonabelian one.

Contribution

It offers a new geometric approach to classify and understand plane sextics with specific singularities, including explicit descriptions and fundamental group computations.

Findings

All fundamental groups are finite.

One fundamental group is nonabelian.

Explicit geometric descriptions of maximal sextics with E7 singularity.

Abstract

We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\bold E_7$ singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian.