A short proof of the Göttsche conjecture

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We prove that for a sufficiently ample line bundle L on a surface S, the number of δ-nodal curves in a general δ-dimensional linear system is given by a universal polynomial of degree δ in the four numbers L²,L.Kₛ,Kₛ² and c₂(S). The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn. We are also able to weaken the ampleness required, from Göttsche's (5δ-1)-very ample to δ-very ample.