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

Author(s): Dalide Pontoni
Journal: Trans. Amer. Math. Soc. 359 (2007), 5419-5448.
MSC (2000): Primary 14N35
Posted: May 17, 2007
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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: 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
Posted: May 17, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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