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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 Vallès, 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.

Schwarzenberger bundles of arbitrary rank on the projective space