Patchworking singular algebraic curves II

In this paper we present various particular versions of the general patchworking procedure for the construction of reduced algebraic curves with prescribed singularities on algebraic surfaces. Among the main examples are a deformation of reducible algebraic curves on reducible algebraic surfaces in the presence of non-transverse intersections of a curve with the singular locus of a surface, and a deformation of curves with multiple components. As an application we deduce a significant asymptotical improvement for the sufficient existence criterion of algebraic curves with arbitrary prescribed singularities in given linear systems on smooth projective algebraic surfaces.