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

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls