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Deformations of Cohen-Macaulay schemes of codimension 2 and nonsingular deformations of space curves. (English) Zbl 0358.14006
This paper, the published version of the author’s dissertation, contains two of the three main results given there - that Cohen-Macaulay schemes of codimension 2 are determinantal, and, for schemes of dimension less than four, are smoothable. To be more precise, the work deals with embedded affine schemes of codimension which locally, and thus globally, have a projective 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 lange to permit one to change the constant and linear terms of each entry in the matrix.
Reviewer: Mary Schaps

##### MSC:
 14D15 Formal methods and deformations in algebraic geometry 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14H45 Special algebraic curves and curves of low genus
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