Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Details Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

2007-10-16

arxiv.org/abs/0710.3072v2

Mathematics

Algebraic Geometry

41 pages; revised version, exposition improved

Scientific paper

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 $\Lambda^k L^{[n]}$.

Affiliated with

Also associated with

No associations

Rating

Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-714417