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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum cohomology of Hilb$ ^2(\mathbb{P}^1 \times\mathbb{P}^1)$ and enumerative applications


Author: Dalide Pontoni
Journal: Trans. Amer. Math. Soc. 359 (2007), 5419-5448
MSC (2000): Primary 14N35
Published electronically: May 17, 2007
MathSciNet review: 2327036
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Abstract: We compute the small quantum cohomology of $ \mathbf{H}=$Hilb $ ^2(\mathbb{P}^1\times\mathbb{P}^1)$ and determine recursively most of the big quantum cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in $ \mathbb{P}^1\times\mathbb{P}^1$. This extends the results obtained by Graber (2001) for Hilb $ ^2(\mathbb{P}^2)$ and hyperelliptic curves in $ \mathbb{P}^2$.


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Additional Information

Dalide Pontoni
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Address at time of publication: Liceo Scientifico Statale “M. Grigoletti”, 33170 Pordenone, Italy
Email: pontoni@iastate.edu, dalide.pontoni@istruzione.it

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04169-4
PII: S 0002-9947(07)04169-4
Received by editor(s): July 30, 2004
Received by editor(s) in revised form: September 22, 2005
Published electronically: May 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.