Schwarzenberger bundles on smooth projective verities

We introduce a generalized notion of Schwarzenberger bundle on the projective space. Associated to this more general definition, we give an ad hoc notion of jumping subspaces of a Steiner bundle on P(n) (which in rank n coincides with the notion of unstable hyperplane introduced by Valles, Ancona and Ottaviani). For the set of jumping hyperplanes, we find a sharp bound for its dimension. We also classify those Steiner bundles whose set of jumping hyperplanes have maximal dimension and prove that they are generalized Schwarzenberger bundles.