Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For instance, if $\mathrm{S}$ is the projective plane, then $\mathrm{Hilb}^2(\mathrm{S})$ is the $\mathbf{P}^2$-bundle $\mathbf{P}(\mathrm{Sym}^2\Omega(1))$ over the dual projective plane. (If this were a toric variety, then $\mathrm{Sym}^2\Omega(1)$ would split.) Nonetheless, the $\mathrm{Hilb}^n(\mathbf{P}^2)$ share quite a few properties with toric varieties (e.g., they are Mori dream spaces, the Hodge numbers satisfy $h^{p,q}=0$ for $p\neq q$).

On the other hand, if $\mathrm{S}$ is the affine plane, then $\mathrm{Hilb}^2(\mathrm{S})$ is toric, since it is isomorphic to the product of the affine plane with the blow up of a quadric cone at its vertex.

For which toric surfaces $\mathrm{S}$ and $n\geqslant 2$ does $\mathrm{Hilb}^n(\mathrm{S})$ admit the structure of a toric variety?