Counting Singular Plane Curves via Hilbert Schemes

Consider the following question. Given a suitable linear series L on a smooth surface S, how many curves in L have a given analytic or topological type of singularity? By “suitable” linear series with respect to a type of singularity, we mean that there are finitely many curves with the singularity in the linear series and their codimension is maximal. Our approach to this question is to express the answer as a Chern number of a vector bundle over a compactification of a space linearizing the condition of having the singularity. For example, the condition of having a cusp along a given tangent direction at a given point is linear in the sense that curves in a linear series spanned by two curves with the condition also have the condition. Thus, the projectivized tangent bundle PT (S) linearizes the condition of having a cusp in S. Note that the closure of the condition of having a cusp in along a given direction is the condition of containing a particular subscheme of S isomorphic to Spec(R/(x, xy, y)) where R is the ring K[[x, y]] and K is the field of definition of S. Generalizing this, the spaces we will use to linearize our conditions will be of the form U(I) = {a ∈ Hilb(S) : a ∼= Spec(R/I)}