## Gromov-Witten invariants of jumping curves

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**360**(2008), 989-1004 Request permission

## 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.## References

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## Additional Information

**Izzet Coskun**- Affiliation: Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 736580
- Email: coskun@math.mit.edu
- Received by editor(s): May 14, 2005
- Received by editor(s) in revised form: February 1, 2006
- Published electronically: May 11, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2346480

Dedicated: A la memoire de Grandmaman Regine