Gromov-Witten invariants of jumping curves

Given a vector bundle on a smooth projective variety , we can define subschemes of the Kontsevich moduli space of genus-zero stable maps parameterizing maps such that the Grothendieck decomposition of 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 . We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on . Our construction is a generalization of jumping lines for vector bundles on . 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.