# tangent bundle of Hilbert schemes of points on a projective surface

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 !

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.