A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points

Abstract

For a smooth projective variety $X$ of dimension $d\ge 5$ over an algebraically closed field $k$ of characteristic zero, it is shown in this paper that the bounded derived category of the Hilbert scheme of three points $X^{[3]}$ admits a semi-orthogonal sequence of length $\left(\genfrac{}{}{0ex}{}{d-3}{2}\right)$. Each subcategory in this sequence is equivalent to the derived category of $X$ and realized as the image of a Fourier-Mukai transform along a Grassmannian bundle $\mathbb{G}$ over $X$ parametrizing planar subschemes in $X^{[3]}$. The main ingredient in the proof is the computation of the normal bundle of $\mathbb{G}$ in $X^{[3]}$. An analogous result for generalized Kummer varieties is deduced at the end.

v2 update (2026-04-03): added Springer disclaimer after abstract; all other content identical to v1 (submitted 2024)

References in corpus (1)

• Derived categories of flips and cubic hypersurfaces