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Gromov-Witten invariants of jumping curves


Author: Izzet Coskun
Journal: Trans. Amer. Math. Soc. 360 (2008), 989-1004
MSC (2000): Primary 14F05, 14J60, 14N10, 14N35
DOI: https://doi.org/10.1090/S0002-9947-07-04284-5
Published electronically: May 11, 2007
MathSciNet review: 2346480
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Abstract: Given a vector bundle $ E$ on a smooth projective variety $ X$, we can define subschemes of the Kontsevich moduli space of genus-zero stable maps $ M_{0,0}(X, \beta)$ parameterizing maps $ f: \mathbb{P}^1 \rightarrow X$ such that the Grothendieck decomposition of $ f^*E$ has a specified splitting type. In this paper, using a ``compactification'' of this locus, we define Gromov-Witten invariants of jumping curves associated to the bundle $ E$. We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on $ \mathbb{P}^4$. Our construction is a generalization of jumping lines for vector bundles on $ \mathbb{P}^n$. Since for the tautological bundle of the Grassmannians the invariants are enumerative, we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls.


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Additional Information

Izzet Coskun
Affiliation: Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: coskun@math.mit.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04284-5
Received by editor(s): May 14, 2005
Received by editor(s) in revised form: February 1, 2006
Published electronically: May 11, 2007
Dedicated: A la memoire de Grandmaman Regine
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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