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 !
Best Answer
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, Algebra Number Theory 10, No. 6, 1173-1190 (2016). ZBL1359.14040.