The Geometry of Equisingular and Equianalytic Families of Curves on a Surface

Introduction The existence of singular algebraic curves with given invariants and given set of singular-ities and, at the same time, the study of (equisingular) families of such curves is a very old, but still attractive and widely open, problem. Already, at the beginning of the 20th century, the foundations were made in the works of Pl ucker, Severi, Segre and Zariski. In the sequel the theory of equisingular families has attracted the continuous attention of algebraic geometers and has found important applications in singularity theory, topology of complex algebraic curves and surfaces and in real algebraic geometry. In this thesis, we concentrate (mainly) on (families of) algebraic curves C on smooth projective surfaces over the complex eld C. Let D be a divisor on. The discriminant in the linear system jDj decomposes into the set of non-reduced or reducible curves and the sets V irr