  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.
