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 $\delta$–nodal curves in a general $\delta$–dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,L.K_S,K_S^2$ and $c_2\left(S\right)$.

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 $\left(5\delta -1\right)$–very ample to $\delta$–very ample.