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.
Keywords
Cite
@article{arxiv.0812.3342,
title = {A syzygetic approach to the smoothability of zero-dimensional schemes},
author = {Daniel Erman and Mauricio Velasco},
journal= {arXiv preprint arXiv:0812.3342},
year = {2012}
}
Comments
22 pages, 1 figure. Corrected typos. Included Macaulay2 code for computations cited in the paper at the end of the laTex version of the document