  A great deal of recent activity has centered on the question of whether, for
a given Hilbert function, there can fail to be a unique minimum set of graded
Betti numbers, and this is closely related to the question of whether the
associated Hilbert scheme is irreducible or not. We give a broad class of
Hilbert functions for which we show that there is no minimum set of graded
Betti numbers, and hence that the associated Hilbert scheme is reducible.
Furthermore, we show that the Weak Lefschetz Property holds for the general
element of one component, while it fails for every element of another
component.
