Unobstructed arithmetically Buchsbaum curves

Abstract

This paper contains the material of a talk that the author gave at the conference Projective curves and Algebraic Geometry at Cognola (Trento, 1988). The author is very grateful to the organizers for their generous hospitality. Let X be a curve in p3. We say that X is unobstructed if the corresponding point of the Hilbert scheme is smooth; otherwise X is called obstructed. A geometric characterization of unobstructedness is not known even for smooth space curves, but several examples of obstructed smooth curves in p3 are known (see for instance [Mu], lSI, IEFI, [Kl], lEI). In the past few years, the subject of arithmetically Buchsbaum curves, as a natural extension of arithmetically Cohen­Macaulay curves, has recieved much attention. In IERI, Ellingsrud proved that arithmetically Cohen­Macaulay curves are unobstructed. However, this is not true for arithmetically Buchsbaum curves (cf.IEFI) and, in IEF 11, Ellia­Fiorentini considered the following problem: PROBLEM 1. To characterize unobstructed arithmetically Buchsbaum curves. In particular, PROBLEM 2. Is any arithmetically Buchsbaum curve of maximal rank unobstructed? The goal of this work is to give sufficient conditions on the numerical character of an arithmetically Buchsbaum curve of maximal rank in order to assure that it is unobstructed (Cf. Theorem 2.1 and Theorem 2.2). The first section is primarily a review of the results about arithmetically Buchsbaum curves needed later. The heart of this paper is § 2 where we prove the main results.