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A Conjectural Generating Function for Numbers of Curves on Surfaces

1998; Springer Science+Business Media; Volume: 196; Issue: 3 Linguagem: Inglês

10.1007/s002200050434

1432-0916

Lothar Göttsche,

Commutative Algebra and Its Applications

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 [V] for the case δ &\le; 6 and Kleiman–Piene [K-P] for the case δ &\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 ℙ2. 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.