Several families of rank-two vector bundles on Hirzebruch
surfaces are shown to consist of all very ample, uniform bundles. Under
suitable numerical assumptions, the projectivization of these bundles,
embedded by their tautological line bundles as linear scrolls, are shown
to correspond to smooth points of components of their Hilbert scheme,
the latter having the expected dimension. If e = 0,1 the scrolls fill up
the entire component of the Hilbert scheme, while for e = 2 the scrolls
exhaust a subvariety of codimension 1.