Cohomology of a tautological bundle on the Hilbert scheme of a surface

We compute the cohomology spaces for the tautological bundle tensor the det erminant bundle on the punctual Hilbert scheme X-[m] of a smooth projective surface X on C. We show that for L vector bundle and A invertible vector b undle on X, if H-q(X, A) = H-q(X, L x A) = 0 for q greater than or equal to 1, then the higher cohomology spaces on X-[m] of the tautological bundle a ssociated to L tensor the determinant bundle associated to A vanish, and th e space of global sections is computed in terms of H-0(A) and H-0(X, L x A) . This result is motivated by the computation of the space of global sectio ns of the determinant bundle on the moduli space of rank 2 semi-stable shea ves on the projective plane, supporting Le Potier's strange duality conject ure on the projective plane.