Abstract:
We prove the conjectures of Yau-Zaslow and Gottsche concerning the number
curves on K3 surfaces. Specifically, let X be a K3 surface and C be a
holomorphic curve in X representing a primitive homology class. We count
the number of curves of geometric genus g with n nodes passing through g
generic points in X in the linear system |C| for any g and n satisfying
C^2=2g+2n-2.
When g=0, this coincides with the enumerative problem studied by Yau
and Zaslow who obtained a conjectural generating function for the numbers.
Recently, Gottsche has generalized their conjecture to arbitrary g in terms
of quasi-modular forms. We prove these formulas using Gromov-Witten
invariants for families, a degeneration argument, and an obstruction bundle
computation. Our methods also apply to P^2 blown up at 9 points where we
show that the ordinary Gromov-Witten invariants of genus g constrained to g
points are also given in terms of quasi-modular forms.