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 singulaties for which our methods suffice include the topological type with local equation xa +yb with b-1 < a < 3b+1. We work out the example of curves with the analytic type of singularity with local equation x2+yn for 1<n<9.
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