Let $S$ be a smooth projective surface. We denote $S^{[n]}$ the Hilbert scheme of artinian subschemes in $S$ of length $n$, which is a smooth projective variety of dimension $2n$ by Fogarty. Let $I\subset S^{[n]}\times S$ be the universal correspondance, and let $p: I\to S^{[n]}$ and $q: I\to S$ be projections. To be precise, the fiber of $p$ over a point in $S^{[n]}$ is the corresponding subscheme of $S$.
We fix the following convention (which is quite common in literature):
Let $F$ be a vector bundle on $S$. $F^{[n]}$ is the locally free coherent sheaf on $S^{[n]}$ defined by $p_*q^*F$.
It seems that the tangent bundle of $S^{[n]}$ coincides with $T_S^{[n]}$ where $T_S$ is the tangent bundle on $S$. The reason why I guess so is the following
The tangent space of $S^{[n]}$ at a point $z\in S^{[n]}$ representing $Z\subset S$ is given by $Hom(I_Z/I_Z^2, \mathcal O_Z)$. In the case where $Z$ is non reduced, $Hom(I_Z/I_Z^2, \mathcal O_Z)\cong H^0(Z,T_{X|Z})$. Thus, $T_{S^{[n]},z}=H^0(Z,T_X|Z)=p_*q^*F|_z$. I do not see if these isomorphisms still hold when $Z$ is a reduced subscheme.
Any comments, responses and references are more than welcome !
