A conjectural generating function for numbers of curves on surfaces

Abstract

I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta \le 6$ and Kleiman-Piene for the case $\delta \le 8$. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in ${\P }_2$. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.

amslatex 13 pages

References in corpus (2)

• Euler number of the compactified Jacobian and multiplicity of rational curves
• Counting curves of any genus on rational ruled surfaces

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