Gromov-Witten invariants of jumping curves

Abstract

Given a vector bundle E E on a smooth projective variety X X , we can define subschemes of the Kontsevich moduli space of genus-zero stable maps M 0 , 0 ( X , β ) M_{0,0}(X, \beta ) parameterizing maps f : P 1 → X f: \mathbb {P}^1 \rightarrow X such that the Grothendieck decomposition of f ∗ E 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 E . We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on P 4 \mathbb {P}^4 . Our construction is a generalization of jumping lines for vector bundles on P n \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.