For a smooth and quasi-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 \({{\,\mathrm{D^b}\,}}(X^{[3]})\) of the Hilbert scheme of three points admits a semi-orthogonal sequence of length \(\left( {\begin{array}{c}d-3\\ 2\end{array}}\right) \) . Each subcategory in this sequence is equivalent to \({{\,\mathrm{D^b}\,}}(X)\) and realized as the image of a Fourier–Mukai transform along a Grassmannian bundle \(\mathbb {G} \rightarrow X\) 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.
