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.

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

References in corpus (3)

• Hilbert schemes of 8 points
• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

Cited by in corpus (7)

• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• Four lectures on secant varieties
• Elementary components of Hilbert schemes
• Filtering free resolutions
• Deformations of zero-dimensional schemes and applications
• Effective divisors on $\ov{\mc{M}}_g$ associated to curves with exceptional secant planes
• The space of generically étale families