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 $\Lambda^k L^{ [n]}$.
Categories: math.AG math.AC math.CO