Counting Singular Plane Curves Via Hilbert Schemes
Abstract
Description
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.
examples added, some results strengthened, some minor errors corrected, some expositional changes
examples added, some results strengthened, some minor errors corrected, some expositional changes