Gromov-Witten invariants of jumping curves

Coskun, Izzet

doi:10.1090/S0002-9947-07-04284-5

Gromov-Witten invariants of jumping curves
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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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