The action of the two-dimensional torus on the plane extends
to an action of the torus on the Hilbert scheme of points in the plane.
The T-graph is the graph whose vertices are the fixed points under the
torus action (corresponding to monomial ideals), where two such vertices are connected by an edge if there exists a one-dimensional torus orbit whose closure contains the corresponding fixed points. I will describe some conditions for the existence of an edge between two given vertices.

This is joint work with Diane Maclaga.