The enumerative geometry of surfaces and modular forms |
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Authors: | Jim Bryan Naichung Conan Leung |
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Affiliation: | Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118 ; School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 |
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Abstract: | Let be a surface, and let be a holomorphic curve in representing a primitive homology class. We count the number of curves of geometric genus with nodes passing through generic points in in the linear system for any and satisfying . When , 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 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 blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus constrained to points are also given in terms of quasi-modular forms. |
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Keywords: | |
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