Some examples of obstructed curves in P³

Summary

In this paper we compute explicit étale neighborhoods of the points in the Hilbert scheme (the universal family of subschemes of ℙ³) corresponding to curves in ℙ³ of a certain type. Most of these curves are obstructed, that is to say, correspond to singularities of the Hilbert scheme. We have several purposes in doing this. One is simply to add to the list of singularities of the Hilbert scheme for which explicit equations have been computed. This is still a very short list which for the most part consists of fairly simple singularities such as two components crossing transversally ([15] 8.6, [14], [7]) or a double structure on a nonsingular variety ([5]). The example of [15] 8.7 seems to be the only explicit singularity known which is more complicated. Our new examples may clarify some of the causes of obstructions, particularly in terms of the cohomology and syzygies of the homogeneous ideal of the curve. A second purpose of the paper is to investigate a question of Sernesi concerning curves of maximal rank. This will be discussed in a moment. But we also simply wish to give an exposition of the deformation theory in terms of which the computation is made so as to make it more accessible to those who study algebraic space curves. We particularly wish to clarify the relationship between the deformations of a space curve (or subscheme) Y ⊂ X = Proj S, the deformations of the ideal sheaf J_y as a coherent sheaf, and the deformations of the homogeneous ideal I(Y) as a homogeneous S–module.