On a discriminant knot group problem of Brieskorn

TL;DR

This paper investigates whether the local fundamental group of the discriminant complement remains constant within the μ-constant stratum of an isolated hypersurface singularity, providing an affirmative answer for certain plane curve germs.

Contribution

It proves the constancy of the local fundamental group for singular plane curve germs of multiplicity at most 3, addressing a longstanding question posed by Brieskorn.

Findings

Confirmed the constancy of the local fundamental group for specific plane curve germs.

Extended understanding of the discriminant complement's topological invariants.

Provided partial affirmative answer to Brieskorn's question in the case of low multiplicity singularities.

Abstract

Quite some time ago, at the singularity conference at Carg\`ese 1972 Brieskorn asked the following question: Is the local fundamental group $\pi_1^{s}(S-D)$ of the discriminant complement inside the semi-universal unfolding $S$ of an isolated hypersurface singularity constant for $s$ in the $\mu$-constant stratum $\Sigma_E$? We review this question and give an affirmative answer in case of singular plane curve germs of multiplicity at most $3$.