Let X a smooth quasi-projective algebraic surface, L a line bundle on X. Let X [n] the Hilbert scheme of n points on X and L [n] the tautological bundle on X [n] naturally associated to the line bundle L on X. We explicitely compute the image Φ(L [n] ) of the tautological bundle L [n] for the Bridgeland-King-Reid equivalence Φ :Sn (X n ). We give, moreover, a characterization of the image Φ(L [n] ⊗· · ·⊗L [n] ) in terms of of the hyperderived spectral sequence E p,q 1 associated to the derived k-fold tensor power of the complex C • L . The study of the Sn-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general k-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of X [n] with values in L [n] ⊗ L [n] and Λ k L [n] .

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland-King-Reid transform. We obtain Brion-Danila's Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert-Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.

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We study the singularities of the isospectral Hilbert scheme B n of n points over a smooth algebraic surface and we prove that they are canonical if n ≤ 5, log-canonical if n ≤ 7 and not log-canonical if n ≥ 9. We describe as well two explicit log-resolutions of B 3 , one crepant and the other S3equivariant.