Notes on diagonals of the product and symmetric variety of a surface

Abstract

Let X be a smooth quasi-projective algebraic surface and let Δn be the big diagonal in the product variety Xn. We study cohomological properties of the ideal sheaves IΔnk and their invariants (IΔnk)Sn by the symmetric group, seen as ideal sheaves over the symmetric variety SnX. In particular we obtain resolutions of the sheaves of invariants (IΔn)Sn for n=3,4 in terms of invariants of sheaves over Xn 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 Δn. 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-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves IΔnk over Xn with respect to very ample line bundles of the form L⊠⋯⊠L and of their sheaves of invariants (IΔnk)Sn on the symmetric variety SnX with respect to very ample line bundles of the form DL.