A short proof of the Göttsche conjecture

We prove that for a sufficiently ample line bundle L on a surface S , the number of ı -nodal curves in a general ı -dimensional linear system is given by a universal polynomial of degree ı in the four numbers L 2 ; L : K S ; K 2 S and c 2 .S /.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 [22] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [8].We are also able to weaken the ampleness required, from Göttsche's .5ı1/-very ample to ı -very ample.