Gromov-witten Invariants of Jumping Curves

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 M0,0(X, β) parameterizing maps f : P1 → 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 P4. Our construction is a generalization of jumping lines for vector bundles on Pn. 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.