Residual equisingularity

Equisingularity at the Normalisation

We look at topological equisingularity of a holomorphic family of reduced mapping germs ft : (C , O) → C over a contractible base T having non-isolated singularities, by means of their normalisations. The relation between surfaces in C with non-isolated singularities and normal surface singularities via the normalisation is a very convenient way of exchanging information between both categories...

Equisingularity and Simultaneous Resolution of Singularities

Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by “dimensionality type,” an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratification by simultaneous resolvability of singularities. The two approaches are known to be equivalent for families of ...

Global equisingularity of families of affine hypersurfaces

The study of the local equisingularity of families of complex analytic hypersurface germs, initiated by Oscar Zariski [EQUI], was accomplished by Bernard Teissier [Te-1]. We refer the reader to e.g. [Za] and [Te-3] for surveys on several types of local equisingularity and their developements. In case of isolated singularities, Teissier introduced the local μ invariants, where μ is the Milnor nu...

Nilpotents, Integral Closure and Equisingularity conditions

Equisingularity in R as Morse Stability in Infinitesimal Calculus

Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in R: For an analytic family ft : (R , 0) → (R, 0), when should it be called an “equisingular deformation”? This amounts to finding a suitable trivialization condition (as strong as possible) and, of course, a criterion. The second is on the Morse stability. We define R∗, which is R “enriched” wit...