Deformations of Cohen-Macaulay Schemes of Codimension 2 and Non-Singular Deformations of Space Curves

Abstract

The following work deals with the deformations of embedded affine schemes of codimension 2, which locally have a resolution of length 2. The cases of immediate interest are curves in 3-space and 0-dimensional schemes in the plane. It is first shown that such a scheme X has a global resolution of length 2. Therefore, by a theorem of Burch, the functions defining the ideal of X can be obtained as the maximal minors of a matrix whose columns generate all the relations among these functions. All flat deformations of X can be obtained simply by deforming this matrix, and this permits the construction of the versal deformation space of X. Finally, for X of dimension 3 or less one can construct non-singular deformations of X by taking a parameter space sufficiently large to permit one to change the constant and linear terms of each entry in the matrix. For X of dimension 4, an example is given in XXX [11] in which the scheme not only has no non-singular deformations, but in fact has no non-isomorphic deformations at all. A brief review of the previous literature will help place these results in perspective. It has long been known, by Bertini's theorem, that the generic deformation of a scheme of codimension 1 is non-singular. As a consequence of the work of Fogarty [5], it was also known that every point, or rather, 0-dimensional scheme in the plane has non-singular deformations; Briancon and Galligo [3] give an explicit construction for such a deformation, splitting the scheme into distinct simple points. This led mathematicians interested in algebraic curves to ask if the generic deformation of a space curve is also non-singular, the question which is settled in this paper. Further direct extension of these results in the case of curves is impossible, since the work of Iarrobina [4] permits the construction of a non-reduced curve in affine 4-space which has no non-singular deformations; however, for reduced curves the question is still open as of this writing.