Abstract:

In 1979 O. Zariski proposed a general theory of equisingularity for algebraic or algebroid hypersurfaces over an algebraically closed field of characteristic zero. This theory is based on the notion of dimensionality type that is defined recursively by considering the discriminants loci of subsequent “generic” projections. Thus the points of dimensionality type 0 are regular points and the singularities of dimensionality type 1, are generic singular points in codimension 1. Zariski proved that the latter ones are isomorphic to the equisingular families of plane curve singularities.

In this talk we give a similar characterization for singularities of dimensionality type 2, i.e. for generic singularities in codimension two. We show that they are isomorphic to equisingular families of surface singularities, with the equisingularity type determined by the 

discriminants of their “generic” projection. Moreover, we show that in this case the generic linear projections are generic (this is still open for dimensionality type greater than 2). Over the field of complex numbers, we show that such families are bi-Lipschitz trivial, by constructing an explicit Lipschitz stratification.

(Based on joint work with L. Paunescu.)