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

TL;DR

This paper constructs a semi-orthogonal sequence in the derived category of the Hilbert scheme of three points on a smooth projective variety, linking it to the derived category of the original variety via Fourier-Mukai transforms.

Contribution

It introduces a new semi-orthogonal decomposition for the derived category of the Hilbert scheme of three points, extending understanding of its structure.

Findings

Semi-orthogonal sequence of length inom{d-3}{2} in the derived category of X^{[3]}

Each subcategory is equivalent to the derived category of X

The proof involves computing the normal bundle of a Grassmannian bundle in X^{[3]}

Abstract

For a smooth projective variety $X$ of dimension $d \geq 5$ 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 $\binom{d-3}{2}$. 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.