Algebro-geometric equisingularity of Zariski

TL;DR

This survey explores Zariski equisingularity, detailing its definition, properties, applications, and methods for constructing equisingular deformations in algebraic and analytic hypersurfaces.

Contribution

It provides a comprehensive overview of Zariski equisingularity, including its relation to other conditions and the canonical stratification of hypersurfaces.

Findings

Zariski equisingularity can be characterized via deformation constructions.

It relates to other equisingularity conditions in algebraic geometry.

Canonical stratification by dimensionality type is significant in the theory.

Abstract

This is a survey on Zariski equisingularity. We recall its definition, main properties, and a variety of applications in Algebraic Geometry and Singularity Theory. In the first part of this survey, we consider Zariski equisingular families of complex analytic or algebraic hypersurfaces. We also discuss how to construct Zariski equisingular deformations. In the second part, we present Zariski equisingularity of hypersurfaces along a nonsingular subvariety and its relation to other equisingularity conditions. We also discuss the canonical stratification of such hypersurfaces given by the dimensionality type.