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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The enumerative geometry of $K3$ surfaces and modular forms
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by Jim Bryan and Naichung Conan Leung PDF
J. Amer. Math. Soc. 13 (2000), 371-410 Request permission


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
  • Email:
  • Naichung Conan Leung
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 610317
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 371-410
  • MSC (2000): Primary 14N35, 53D45, 14J28
  • DOI:
  • MathSciNet review: 1750955