Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points

For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier--Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of $X^{[n]}$ is always flat over $X$, which implies that the Fourier--Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.