Equivariant homology of Hilb and torus stable curves

Question

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

Answer 1

It doesn't work as simply as it does for $G/B$, because there aren't finitely many $T$-invariant curves. Instead they will come in families whose unions are things like products of projective spaces. Although there will be a larger torus acting on each such family which has finitely many $T$-invariant curves, it won't extend to an action on the whole variety.

If you can work out the $T$-equivariant cohomology of the components of the fixed point sets for all codimension $1$ subtori of $T$, then you can take the intersection of all those things inside $H_T(X^T)$ just as you would if there were finitely many $T$-invariant curves. This has been worked out for $Hilb_n(C^2)$, and more generally the Hilbert scheme of a smooth toric surface, in Evain, "The Chow ring of punctual Hilbert schemes of toric surfaces".