A syzygetic approach to the smoothability of zero-dimensional schemes

Abstract

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.Comment: 22 pages, 1 figure. Corrected typos. Included Macaulay2 code for computations cited in the paper at the end of the laTex version of the documen

Some elementary components of the Hilbert scheme of points

“…By Corollary 7.6 the image of the associated map F : U → Hilb 13 Remark 8.1. The notebook also includes an example of a generic distinguished ideal with Hilbert function (1,5,3,4, 0) that is not efficient. Therefore, efficiency is not necessary for genericity.…”

“…Consequently, ϑ-efficiency is not a necessary condition for genericity, as we noted in Remark 3.6. (1,6,21,10,15,0), shape (6,3,3,4). The details of this example are presented in the notebook case (1, 6, 21, 10, 15, 0).nb.…”

“…The present paper is but one small contribution to the voluminous, diverse, and rapidly increasing literature on components of Hilbert schemes of points; for example, see [1], [2], [3], [4], and the references contained therein.…”

Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

“…Schemes that are in H r (X) are called smoothable in X (because there exists a flat irreducible deformation to a smooth scheme). It is an interesting and non-trivial problem to determine when Hilb r (X) = H r (X), and to identify the schemes that are in H r (X) if the equality does not hold-see, e.g., [10], [23] and references therein. Recall that Lemma 1.1.2 states that for d ≥ d 0 the following equality of linear spans holds:…”

Smoothable zero dimensional schemes and special projections of algebraic varieties

“…Then is smoothable if and only if the corresponding point lies in . Whether a given R is smoothable is a difficult question, see . It is connected with the search for equations of secant varieties .…”