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

8 pages. Published version

References in corpus (1)

• Curves on K3 surfaces and modular forms

Cited by in corpus (19)

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• Tautological integrals on curvilinear Hilbert schemes
• Enumeration of curves with one singular point
• Virtual ${\chi }_{-y}$-genera of Quot schemes on surfaces
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• The Di Francesco-Itzykson-Göttsche Conjectures for Node Polynomials of ${\mathbb{P}}^2$
• Enumeration of rational curves in a moving family of ${\mathbb{P}}^2$
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