Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The enumerative geometry of K3 surfaces and modular forms   We prove the conjectures of Yau-Zaslow and Gottsche concerning the number
curves on K3 surfaces. Specifically, let X be a K3 surface and 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 |C| for any g and n satisfying C^2=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, Gottsche 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 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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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph