The spine of the \(T\)-graph of the Hilbert scheme of points in the plane

The torus \(T\) of projective space also acts on the Hilbert scheme of subschemes of projective space. The \(T\)-graph of the Hilbert scheme has vertices the fixed points of this action, and edges connecting pairs of fixed points in the closure of a one-dimensional orbit. In general this graph depends on the underlying field. We construct a subgraph, which we call the spine, of the \(T\)-graph of \(Hilb^m( A^2)\) that is independent of the choice of infinite field. For certain edges in the spine we also give a description of the tropical ideal, in the sense of tropical scheme theory, of a general ideal in the edge. This gives a more refined understanding of these edges, and of the tropical stratification of the Hilbert scheme.