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

Abstract

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 ${\mathcal{I}}_{{\Delta }_n}^k$ and their invariants $({\mathcal{I}}_{{\Delta }_n}^k{)}^{{\mathfrak{S}}_n}$ by the symmetric group, seen as ideal sheaves over the symmetric variety $S^{n}X$. 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-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves ${\mathcal{I}}_{{\Delta }_n}^k$ over $X^n$ with respect to very ample line bundles of the form $L⊠\cdots ⊠L$ and of their sheaves of invariants $({\mathcal{I}}_{{\Delta }_n}^k{)}^{{\mathfrak{S}}_n}$ on the symmetric variety $S^{n}X$ with respect to very ample line bundles of the form ${\mathcal{D}}_L$.

arXiv comment34 pages, exposition improved