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
The technique is a study of Hilbert schemes of points on curves on a surface,
using the BPS calculus of Pandharipande and the third author [J. Amer. Math. Soc.
23 (2010) 267–297] and the computation of tautological integrals on Hilbert
schemes by Ellingsrud, Göttsche and Lehn [J. Algebraic Geom. 10 (2001)
We are also able to weaken the ampleness required, from Göttsche’s (5δ−1)–very
ample to δ–very ample.