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Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants $(\mathcal{I}^k_{\Delta_n})^{\mathfrak{S}_n}$ by the symmetric group, seen as ideal sheaves over the symmetric variety $S^nX$. In particular we obtain resolutions of the sheaves of invariants $(\mathcal{I}_{\Delta_n})^{\mathfrak{S}_n}$ for $n = 3,4$ in terms of invariants of sheaves over $X^n$ whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal $\Delta_n$. We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with $3$ and $4$ points and we give universal formulas for their Euler-Poincar\'e characteristic. Finally, we obtain upper bounds for the regularity of the sheaves $\mathcal{I}^k_{\Delta_n}$ over $X^n$ with respect to very ample line bundles of the form $L \boxtimes \cdots \boxtimes L$ and of their sheaves of invariants $( \mathcal{I}^k_{\Delta_n})^{\mathfrak{S}_n}$ on the symmetric variety $S^nX$ with respect to very ample line bundles of the form $\mathcal{D}_L$., Comment: 34 pages, exposition improved

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 $n \leq 5$, log-canonical if $n \leq 7$ and not log-canonical if $n \geq 9$. We describe as well two explicit log-resolutions of $B^3$, one crepant and the other $\mathfrak{S}_3$-equivariant., Comment: 12 pages

We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle $L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smooth quasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$ be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbf{\Phi}(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, which operate locally as higher order restrictions to pairwise diagonals. We use this description and the natural filtration on $(S^k V_L)^{\mathfrak{S}_n}$ induced by the decomposition in direct sum, to obtain, for $n =2$ or $k \leq 4$, a finite decreasing filtration $\mathcal{W}^\bullet$ on the direct image $\mu_*(S^k L^{[n]})$ for the Hilbert-Chow morphism whose graded sheaves we control completely. As a consequence of this structural result, we obtain a chain of cohomological consequences, like a spectral sequence abutting to the cohomology of symmetric powers $S^k L^{[n]}$, an effective vanishing theorem for the cohomology of symmetric powers $S^k L^{[n]} \otimes \mathcal{D}_A$ twisted by the determinant, in presence of adequate positivity hypothesis on $L$ and $A$, as well as universal formulas for their Euler-Poincar\'e characteristic., Comment: pdflatex, 70 pages

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 $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex $\comp{\mc{C}}_L$. The study of the $\perm_n$-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 $L^{[n]} \tens L^{[n]}$ and $\Lambda^k L^{[n]}$., Comment: 41 pages; revised version, exposition improved