Geometry of equisingular families of plane algebraic curves

Let V be the set of irreducible plane curves of a given degree d with singular points of given topological or analytic types (equisingular stratum), defined over an algebraically closed field of characteristic zero. We show that if the number of singular points is bounded from above by a quadratic function of d, then V is a smooth irreducible variety, which locally is the transversal intersection of germs of equisingular strata of individual singular points. This estimate gives an optimal asymptotics with respect to the exponent of d. Under the same condition we get the independence of simultaneous deformations of plane curve singular points as well.