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1 June 2001 On the quantum cohomology of a symmetric product of an algebraic curve
Aaron Bertram, Michael Thaddeus
Duke Math. J. 108(2): 329-362 (1 June 2001). DOI: 10.1215/S0012-7094-01-10825-9


The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example, as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [(3/4)g, g−1). Otherwise, they still give a formula modulo third-order terms. Explicit generators and relations are also given unless d is in [(4/5)g−3/5, g−1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically, a formula of J. Harris and L. Tu for the Chern numbers of determinantal varieties. The case of d=g−1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with A. Givental's work is also explained.


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Aaron Bertram. Michael Thaddeus. "On the quantum cohomology of a symmetric product of an algebraic curve." Duke Math. J. 108 (2) 329 - 362, 1 June 2001.


Published: 1 June 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1050.14052
MathSciNet: MR1833394
Digital Object Identifier: 10.1215/S0012-7094-01-10825-9

Primary: 14N35
Secondary: 14H51

Rights: Copyright © 2001 Duke University Press


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Vol.108 • No. 2 • 1 June 2001
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