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- W2112291245 abstract "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 $Lambda^k L^{[n]}$." @default.
- W2112291245 created "2016-06-24" @default.
- W2112291245 creator A5062409487 @default.
- W2112291245 date "2009-11-01" @default.
- W2112291245 modified "2023-10-16" @default.
- W2112291245 title "Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles" @default.
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- W2112291245 doi "https://doi.org/10.1215/00127094-2009-050" @default.
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