On algebraic representatives of homeomorphism types of analytic hypersurface germs

TL;DR

This paper proves that for complex analytic hypersurface germs with 1-dimensional singular sets, there exist algebraic hypersurface germs sharing the same topological type, extending previous results limited to isolated singularities.

Contribution

It establishes the existence of algebraic representatives for topological types of hypersurface germs with 1-dimensional singular sets, broadening the scope beyond isolated singularities.

Findings

Confirmed algebraic representatives exist for hypersurface germs with 1-dimensional singular sets.

Extended the class of hypersurface germs for which topological types can be realized algebraically.

Provided affirmative answer to a question posed by B. Teissier regarding topological equivalence.

Abstract

A question of B. Teissier, inspired by a previous problem of R. Thom, asks whether for any germ of complex analytic hypersurface there exists a germ of complex algebraic hypersurface with the same topological type. Up to now only the case of germs with an isolated singularity is kwown. In this article we answer in the affirmative the case 1-dimensional singular set, also for embedded topological types.