The enumerative geometry of $K3$ surfaces and modular forms

Let $X$ be a $K3$ surface, and let $C$ 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 $\left\vert C\right\vert$ for any $g$ and $n$ satisfying $C\cdot C=2g+2n-2$.

When $g=0$, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche 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 $\mathbf{P}^{2}$ 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.