A short proof of the Göttsche conjecture

Abstract

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 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 [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) 81–100]. We are also able to weaken the ampleness required, from Göttsche’s ( 5 δ − 1 ) –very ample to δ –very ample.