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

For a smooth and quasi-projective variety X of dimension d ≥ 5 d ≥ 5 over an algebraically closed field k of characteristic zero, it is shown in this paper that the bounded derived category \,Dᵇ\,(X[3]) D b ( X [ 3 ] ) of the Hilbert scheme of three points admits a semi-orthogonal sequence of length ( arraycd-3\\ 2array) d - 3 2 . Each subcategory in this sequence is equivalent to \,Dᵇ\,(X) D b ( X ) and realized as the image of a Fourier–Mukai transform along a Grassmannian bundle G → 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 G G in X[3] X [ 3 ] . An analogous result for generalized Kummer varieties is deduced at the end.