Tetrahedral Curves

Abstract

A tetrahedral curve is a space curve whose defining ideal is an intersection of powers of monomial prime ideals of height two. It is supported on a tetrahedral configuration of lines. Schwartau described when certain such curves are ACM, namely he restricted to curves supported on a certain four of the six lines. We consider the general situation. We first show that starting with an arbitrary tetrahedral curve, there is a particular reduction that produces a smaller tetrahedral curve and preserves the even liaison class. We call the curves that are minimal with respect to this reduction S-minimal curves. Given a tetrahedral curve, we describe a simple algorithm (involving only integers) that computes the S-minimal curve of the corresponding even liaison class; in the process it determines if the original curve is arithmetically Cohen-Macaulay or not. We also describe the minimal free resolution of an S-minimal curve, using the theory of cellular resolutions. This resolution is always linear. This result allows us to classify the arithmetically Buchsbaum, non-ACM tetrahedral curves. More importantly, it allows us to conclude that an S-minimal curve is minimal in its even liaison class; that is, the whole even liaison class can be built up from the S-minimal curve. Finally, we show that there is a large set of S-minimal curves such that each curve corresponds to a smooth point of a component of the Hilbert scheme and that this component has the expected dimension.