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 x a + y b with ⩽ a⩽3 b. We work out the example of curves with the analytic type of singularity with local equation x 2+ y n for 1< n<9.

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