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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Gromov-Witten invariants of jumping curves
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by Izzet Coskun PDF
Trans. Amer. Math. Soc. 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.
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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

  • Dedicated: A la memoire de Grandmaman Regine
  • © 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