Generators for the ideal of an arithmetically Buchsbaum curve

Abstract

This paper continues the study of arithmetically Buchsbaum curves in P 3 by focusing on their hyperplane sections. Here we are especially concerned with the relation between the minimal generators of the ideal of such a curve C and those of the ideal of its hyperplane section C ∩ H. By analyzing the structure of these ideals (which are closely related to the Hartshorne—Rao module M( C )) we are able to give two inequalities for the number of minimal generators of the ideal of C . (The second is originally due to M. Amasaki; we give a new and shorter proof of it.) We also show how the second inequality immediately gives Amasaki's bound for the least degree of a surface containing C . These considerations yield a number of results about C , for instance bounds on the index of speciality e( C ). Finally, we apply these techniques to classify the arithmetically Cohen—Macaulay and Buchsbaum curves on a smooth cubic surface.