A conjectural generating function for numbers of curves on surfaces
Abstract
I give a conjectural generating function for the numbers of
δ
-nodal curves in a linear system of dimension
δ
on an algebraic surface. It reproduces the results of Vainsencher for the case
δ≤6
and Kleiman-Piene for the case
δ≤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
¶
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.