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

TL;DR

This paper investigates the cohomological properties and invariants of ideal sheaves of diagonals in product surfaces, providing resolutions, relations to Hilbert schemes, and bounds on regularity.

Contribution

It offers explicit resolutions of invariant sheaves for n=3,4, relates diagonal ideals to Hilbert schemes via Bridgeland-King-Reid, and establishes bounds on their regularity.

Findings

Resolutions of invariant sheaves for n=3,4

Connections between diagonal ideals and Hilbert schemes

Upper bounds for regularity of diagonal sheaves

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 $\mathcal{D}_L$.