Smoothing projectively Cohen-Macaulay space curves (Q798397)

A (possibly singular) curve in \({\mathbb{P}}^ 3_{{\mathbb{C}}}\) is smoothable if there exists a (flat) deformation to a smooth curve. This paper gives a necessary and sufficient condition for a projectively CM curve to be smoothable. Let Y be a projectively CM curve in \({\mathbb{P}}^ 3\), meaning the homogeneous coordinate ring of Y is Cohen-Macaulay. Let \(r_ 1\leq r_ 2\leq...\leq r_ k\) be the degrees of the generators of the homogeneous ideal I(Y) and \(s_ 1\leq s_ 2\leq...\leq s_{k-1}\) the degrees of the relations between the generators. Then Y is smoothable if and only if \(s_ n\geq r_{n+2}\) for \(n=1,2,...,k-2\). - The theorem provides several examples which allow us to examine the boundary between smoothable and nonsmoothable curves. For example, it reproves the fact that the union of a degree d plane curve and a degree e plane curve, \(d\leq e,\) meeting at d points in \({\mathbb{P}}^ 3\), is smoothable if and only if \(e\leq d+2.\) As another example, let \(Y_ 1\) be the twisted cubic in \({\mathbb{P}}^ 3\), and let \(Y_ 2\) be a plane quartic meeting \(Y_ 1\) transversally at three points. Then \(Y=Y_ 1\cup Y_ 2\) is a smoothable curve. If \(Y_ 2\) is replaced by a plane quintic meeting \(Y_ 1\) at three points, then Y is nonsmoothable.