  We consider the question of which zero-dimensional schemes deform to a
collection of distinct points; equivalently, we ask which Artinian k-algebras
deform to a product of fields. We introduce a syzygetic invariant which sheds
light on this question for zero-dimensional schemes of regularity two. This
invariant imposes obstructions for smoothability in general, and it completely
answers the question of smoothability for certain zero-dimensional schemes of
low degree. The tools of this paper also lead to other results about Hilbert
schemes of points, including a characterization of nonsmoothable
zero-dimensional schemes of minimal degree in every embedding dimension d\geq
4.
