Varchenko showed that if a complex or real analytic set is Zariski equisingular along an affine subspace T then it is locally topologically trivial along T. Using Whitney interpolation we construct explicitly a trivialization of a Zariski equisingular set that is moreover analytic on real analytic arcs and (complex resp. real) analytic with respect to the parameter space T.

Then, given an algebraic set or a germ of an analytic set, we construct

its stratification that fibers this set, locally along each stratum, into analytic submanifolds with a strong continuity of tangent spaces, analogous to Verdier's condition (w). This shows Whitney's fibering conjecture.

We give applications to the general position on singular spaces and to the equisingularity of analytic function germs.

Coauthor: Laurentiu Paunescu