The fundamental group of a generalized trigonal curve

TL;DR

This paper introduces a modified method for computing the fundamental group of trigonal curves with improper fibers and applies it to classify and analyze the fundamental groups of certain sextic curves with specific singularities.

Contribution

It develops a new modification of the Zariski–van Kampen approach and applies it to classify fundamental groups of irreducible maximizing simple sextics with a type D singularity.

Findings

Computed fundamental groups for all irreducible maximizing simple sextics with a type D singularity.

Listed deformation families of these sextics.

Enhanced understanding of the topology of trigonal curves with improper fibers.

Abstract

We develop a modification of the Zariski--van Kampen approach for the computation of the fundamental group of a trigonal curve with improper fibers. As an application, we list the deformation families and compute the fundamental groups of all irreducible maximizing simple sextics with a type $\bold D$ singular point.