A semi-orthogonal sequence in the derived category of the Hilbert scheme of three points

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