I don't know about schemes and every definition of a Hilbert scheme (quite naturally!) involves schemes. But, the Hilbert scheme of points on a complex surface is known to be smooth (Fogarty). So is there a concrete description of it as a complex manifold? (For instance in the case of n=2 it is a blowup of XxX along the diagonal)



Hilbert scheme of points on a complex surface
Given a codimension $d$ ideal $I$ in $R = {\mathbb C}[x,y]$, the quotient ring $R/I$ can be thought of as a $d$-dimensional vector space with actions of two commuting operators $x,y$ and a "cyclic" vector $1$ that generates it as an $R$-module.


Consequently, if your surface is the plane you can think of the Hilbert scheme as the space of pairs of commuting matrices on $d$-space, but then take the (open) set in there of pairs $(x,y)$ that admit a cyclic vector, and then divide that variety by the conjugation action of $GL(d)$. To see smoothness, you might first show this open set is smooth, then that the $GL(d)$ action is free and proper.


All this is spelled out in Nakajima's book (which I'm guessing is the one Andrea Ferretti meant to reference), except I think he shows smoothness by analyzing the tangent spaces.


