Enumeration of singular algebraic curves

Abstract

We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on the intersection theory combined with liftings and degenerations. The procedure computes the homology class in question whenever a given singularity type is defined. Our method does not require knowledge of all the possible deformations of a given singularity, as it was in previously known procedures.