  We give a notion of "combinatorial proximity" among strongly stable ideals in
a given polynomial ring with a fixed Hilbert polynomial. We show that this
notion guarantees "geometric proximity" of the corresponding points in the
Hilbert scheme. We define a graph whose vertices correspond to strongly stable
ideals and whose edges correspond to pairs of adjacent ideals. Every term order
induces an orientation of the edges of the graph. This directed graph describes
the behavior of the points of the Hilbert scheme under Gr\"obner degenerations
with respect to the given term order.
  Then, we introduce a polyhedral fan that we call Gr\"obner fan of the Hilbert
scheme. Each cone of maximal dimension corresponds to a different directed
graph induced by a term order. This fan encodes several properties of the
Hilbert scheme. We use these tools to present a new proof of the connectedness
of the Hilbert scheme. Finally, we improve the technique introduced in the
paper "Double-generic initial ideal and Hilbert scheme" by Bertone, Cioffi and
Roggero to give a lower bound on the number of irreducible components of the
Hilbert scheme.
