Families of reduced zero-dimensional schemes

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.