The enumerative geometry of 𝐾3 surfaces and modular forms

Let X X be a K 3 K3 surface, and let C C be a holomorphic curve in X X representing a primitive homology class. We count the number of curves of geometric genus g g with n n nodes passing through g g generic points in X X in the linear system | C | | C | for any g g and n n satisfying C ⋅ C = 2 g + 2 n − 2 C· C=2g+2n-2 . When g = 0 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 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 P² blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g g constrained to