Counting singular plane curves via Hilbert schemes

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 x(a) + y(b) with less than or equal toaless than or equal to3b. We work out the example of curves with the analytic type of singularity with local equation x(2) + y(n) for 1 < n < 9. (C) 2002 Elsevier Science (USA). All rights reserved.