A syzygetic approach to the smoothability of zero-dimensional schemes

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⩾4d⩾4.