The enumerative geometry of 𝐾3 surfaces and modular forms

Bryan, Jim¹ ; Leung, Naichung²

¹ Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118
² School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Journal of the American Mathematical Society, Tome 13 (2000) no. 2, pp. 371-410

Résumé

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 $\left | C\right |$ 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.

Détail

DOI : 10.1090/S0894-0347-00-00326-X

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