Real plane algebraic curves with prescribed singularities

Abstract

We give a survey on the known results about the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer d and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree d having singular points of the given type as its only singularities. The set of all such curves is a quasiprojective variety, which we call an equisingular family, denoted by ESF. We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and T-smoothness (i.e., being smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we pay special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for T-smooth equisingular families if the degree goes to infinity.