Gromov-Witten invariants of jumping curves

Given a vector bundle E on a smooth projective variety X, we can de. ne subschemes of the Kontsevich moduli space of genus-zero stable maps M-0,M-0(X,beta) parameterizing maps f : P-1 -> X such that the Grothendieck decomposition of f* E has a specified splitting type. In this paper, using a "compactification" of this locus, we de. ne 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 P-4. Our construction is a generalization of jumping lines for vector bundles on 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.