T-graphs of Hilbert schemes of points in the plane

The Hilbert scheme of n points in the plane is a smooth algebraic variety of dimension 2n parametrizing configurations of n points in the plane. Such a configuration of n points in the plane corresponds to an ideal in the polynomial ring k[x,y] of k-vector space dimension n. The natural action of the 2-dimensional torus T on the plane induces an action of T on the Hilbert scheme of such ideals. One can associate to this action a hypergraph whose vertices correspond to the fixed points and edges to 1-dimensional T-orbits. In fact, the fixed points of the torus action correspond to monomial ideals, which in turn correspond to partitions of n, making this graph into a purely combinatorial object. This talk will be of an experimental nature, with a lot of examples.