On irreducible sextics with non-abelian fundamental group

TL;DR

This paper computes the fundamental groups of certain irreducible plane sextic curves with simple singularities, revealing all such groups with dihedral quotient D_{10} are finite, including some of notably large order.

Contribution

It provides a complete calculation of fundamental groups for a class of irreducible sextic curves with specific singularities and dihedral quotients, expanding understanding of their topological properties.

Findings

All computed groups are finite.

Two groups have large orders: 960 and 21600.

The groups admit a dihedral quotient D_{10}.

Abstract

We calculate the fundamental groups $\pi=\pi_1(P^2\setminus B)$ for all irreducible plane sextics $B\subset\P^2$ with simple singularities for which $\pi$ is known to admit a dihedral quotient $D_{10}$. All groups found are shown to be finite, two of them being of large order: 960 and 21600.