2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60436We 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 singulaties for which our methods suffice include the topological type with local equation x^a +y^b with b-1 < a < 3b+1. We work out the example of curves with the analytic type of singularity with local equation x^2+y^n for 1<n<9.examples added, some results strengthened, some minor errors corrected, some expositional changesAlgebraic Geometry14N10text