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 $M_{0,0}(X, \beta)$ 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.