A syzygetic approach to the smoothability of zero-dimensional schemes

Details A syzygetic approach to the smoothability of zero-dimensional schemes A syzygetic approach to the smoothability of zero-dimensional schemes

2008-12-17

arxiv.org/abs/0812.3342v4

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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