Counting singular plane curves via Hilbert schemes

Abstract We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singularities for which our results suffice include the topological type with local equation xa+yb with ⩽a⩽3b. We work out the example of curves with the analytic type of singularity with local equation x2+yn for 1