Hilbert squares: derived categories and deformations

For a smooth projective variety $X$ with exceptional structure sheaf, and $\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X) \to\mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^2X)$ 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 $\operatorname{Hilb}^2X$ 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.