A short proof of the G\"ottsche conjecture

TL;DR

This paper proves that the count of δ-nodal curves in a linear system on a surface can be expressed by a universal polynomial, using Hilbert schemes and BPS calculus, and relaxes ampleness conditions.

Contribution

It provides a new proof of the G"ottsche conjecture, establishing universality of nodal curve counts with weaker ampleness assumptions.

Findings

Count of δ-nodal curves given by a universal polynomial.

Reduced ampleness requirement from (5δ-1)-very ample to δ-very ample.

Utilized Hilbert schemes and BPS calculus for the proof.

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 $c_2(S)$. 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, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample.