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 !



tangent bundle of Hilbert schemes of points on a projective surface
The tangent bundle on $S^{[n]}$ is not quite isomorphic to $(T\_S)^{[n]}$, rather by Theorem B of Stapleton's paper listed below there is an injection of the former into the latter.


*Stapleton, David*, [**Geometry and stability of tautological bundles on Hilbert schemes of points**](http://dx.doi.org/10.2140/ant.2016.10.1173), Algebra Number Theory 10, No. 6, 1173-1190 (2016). [ZBL1359.14040](https://zbmath.org/?q=an:1359.14040).


