Equisingular Families of Curves

We are ready to accomplish our main task, that is to answer the two following questions concerning equisingular families (ESF) of curves. whether a family of algebraic curves with a prescribed collection of singularities form a nonempty, T-smooth (i.e. smooth of expected dimension), irreducible stratum in the discriminant in a given linear system |D| on a smooth algebraic surface \({\varSigma }\), and what is the local structure of the discriminant in a neighborhood of the above stratum, in particular, when is a family deformation complete (i.e. when are any simultaneous local deformations of the curve singularities induced by the given linear system).