Hilbert squares: derived categories and deformations

For a smooth projective variety X with exceptional structure sheaf, and \({X}^{[2]}\) the Hilbert scheme of two points on X, we show that the Fourier–Mukai functor \({{\,\mathrm{\mathbf {D}}\,}}^\mathrm {b}(X) \rightarrow {{\,\mathrm{\mathbf {D}}\,}}^\mathrm {b}({X}^{[2]})\) induced by the universal ideal sheaf is fully faithful, provided the dimension of X is at least 2. This fully faithfulness allows us to construct a spectral sequence relating the deformation theories of X and \({X}^{[2]}\) and to show that it degenerates at the second page, giving a Hochschild–Kostant–Rosenberg-type filtration on the Hochschild cohomology of X. These results generalise known results for surfaces due to Krug–Sosna, Fantechi and Hitchin. Finally, as a by-product, we discover the following surprising phenomenon: for a smooth projective variety of dimension at least 3 with exceptional structure sheaf, it is rigid if and only if its Hilbert scheme of two points is rigid. This last fact contrasts drastically to the surface case: non-commutative deformations of a surface contribute to commutative deformations of its Hilbert square.