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