A note on the Koszul homology of ordinary singularities

Abstract

A great deal has by now been said about syzygies of an ideal in a noetherian ring. The theory is particularly enhanced in the case of a homogeneous ideal in a polYnomial ring (over a field), where there is a pervasive invariance not to mention the concrete geometry waiting in the comer. However, it is often the case when one needs precise information on the length of syzygies and the degrees of the generating cycles. I have personally found it very instructive, if not terribly important, to juggle with such computations. Recently, D. Buchsbaum and D. Eisenbud have together with some students engaged in developing a computer program to quickly recognize syzygies. Not seldom does happen such a computation to shed some light back into the theory. It is my purpose in this note to give one instance of this phenomenon. Explicitly, I am concerned with classifying projective plane curves by means of invariants of the corresponding "gradient ideals". A complete classification is obtained for cubic curves. I take this opportunity to thank D. Lazard for helpftil ideas through a pleasant and enlightening correspondence.