Let X be a proper algebraic scheme over an algebraically closed field. We assume that a
torus T acts on X such that the action has isolated fixed points. The T-graph of X can
be defined using the fixed points and the one-dimensional orbits of the T-action. If the
upper Borel subgroup of the general linear group with maximal torus T acts on X, then
we can define a second graph associated to X, called the A-graph of X. We prove that
the A-graph of X is connected if and only if X is connected. We use this result to give
proof of Hartshorne’s theorem on the connectedness of the Hilbert scheme in the case of
d points in n-dimensional projective space.
.