  We compute the cohomology spaces for the tautological bundle tensor the
determinant bundle on the punctual Hilbert scheme H of subschemes of length n
of a smooth projective surface X. We show that for L and A invertible vector
bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$,
$w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology
spaces on H of the tautological bundle associated to L tensor the determinant
bundle associated to A vanish, and the space of global sections is computed in
terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by the
computation of the space of global sections of the determinant bundle on the
moduli space of rank 2 semi-stable sheaves on the projective plane, supporting
Le Potier's Strange duality conjecture on the projective plane.
