  We consider tautological bundles and their exterior and symmetric powers on
the Quot scheme over the projective line. We prove and conjecture several
statements regarding the vanishing of their higher cohomology, and we describe
their spaces of global sections via tautological constructions. To this end, we
make use of the embedding of the Quot scheme as an explicit local complete
intersection in the product of two Grassmannians, studied by Str{\o}mme. This
allows us to construct resolutions with vanishing cohomology for the
tautological bundles and their exterior and symmetric powers.

Algebraic Geometry