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The enumerative geometry of $K3$ surfaces and modular forms

Authors: Jim Bryan and Naichung Conan Leung
Journal: J. Amer. Math. Soc. 13 (2000), 371-410
MSC (2000): Primary 14N35, 53D45, 14J28
Published electronically: January 31, 2000
MathSciNet review: 1750955
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Abstract: 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.

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Additional Information

Jim Bryan
Affiliation: Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118
ORCID: 0000-0003-2541-5678

Naichung Conan Leung
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID: 610317

Received by editor(s): January 5, 1998
Received by editor(s) in revised form: October 18, 1999
Published electronically: January 31, 2000
Additional Notes: The first author is supported by a Sloan Foundation Fellowship and NSF grant DMS-9802612 and the second author is supported by NSF grant DMS-9626689.
Article copyright: © Copyright 2000 American Mathematical Society