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Counting Singular Plane Curves Via Hilbert Schemes

Heather Russell
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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 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.

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Heather Russell. 2001-01-14. Counting Singular Plane Curves Via Hilbert Schemes. https://arxiv.org/abs/math/0011214

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