Zariski's dimensionality type of singularities. case of dimensionality type 2

TL;DR

This paper advances the understanding of hypersurface singularities by fully analyzing Zariski's equisingularity theory for dimensionality type 2, focusing on surface singularities in three-dimensional space.

Contribution

It completes the classification of Zariski's equisingularity for dimensionality type 2, extending the theory beyond the previously understood type 1 cases.

Findings

Complete classification of Zariski's equisingular families of surface singularities of type 2.

Extension of equisingularity theory to non-isolated surface singularities.

Deeper understanding of the structure of hypersurface singularities in three-dimensional space.

Abstract

In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive "generic" corank 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.