The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. Combinatorially, this produces a copy of the Hasse diagram of Bruhat order inside $G/B$, with edges labeled by appropriate roots. The torus invariant curves can be given explicitly as the images of certain $SL_2$'s inside $G/B$. Is there a similarly explicit construction of the torus invariant curves inside the Hilbert scheme of $n$ points in $\mathbb{C}^2$? If so, does one obtain the same description of equivariant homology via localization (with dominance order replacing Bruhat order)?

Edit: Thanks to David Speyer for the explicit construction of a torus invariant curve here: Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane

The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. Combinatorially, this produces a copy of the Hasse diagram of Bruhat order inside $G/B$, with edges labeled by appropriate roots. The torus invariant curves can be given explicitly as the images of certain $SL_2$'s inside $G/B$. Is there a similarly explicit construction of the torus invariant curves inside the Hilbert scheme of $n$ points in $\mathbb{C}^2$? If so, does one obtain the same description of equivariant homology via localization (with dominance order replacing Bruhat order)?

Edit: Thanks to David Speyer for the explicit construction of a torus invariant curve here: Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane