Mathematics – Algebraic Geometry
Scientific paper
1997-11-11
Mathematics
Algebraic Geometry
amslatex 13 pages
Scientific paper
10.1007/s002200050434
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.
No associations
LandOfFree
A conjectural generating function for numbers of curves on surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A conjectural generating function for numbers of curves on surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A conjectural generating function for numbers of curves on surfaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-200634