On liaison, arithmetical Buchsbaum curves and monomial curves in [formula omitted

Abstract

In the context of liaison, by [ 181 and [ 191, for curves in Ip3, the property of being the zero scheme of a section of a rank two bundle on iP3 is connected to that of being ideally the intersection of three surfaces. An immediate consequence of this property is that the homogeneous ideal I(C) of a curve.C c P3 is generated by precisely three elements if and only if C is ideally the intersection of three suraces and C is arithmetically Cohen-Macaulay (non-complete intersection). Furthermore, the characterization of the equivalence classes, defined by liaison among curves in P3, yields the well-known statement that a curve C is in the liaison class of a complete intersection if and only if C is arithmetically Cohen-Macaulay (see, e.g., [2+ 10, 13, or 141). It seems that A. Cayley (see [7, p. 1521) in 1847 was the first who posed the problem to describe this liaison class of a complete intersection. Now, in this paper we will begin to investigate the next simple case; that is, the liaison classes which are characterized by a finite-dimensional vector space of dimension >l. Using the theory of Buchsbaum rings (see Section 2) this means we will study liaison among arithmetical Buchsbaum curves in P3. First, from the point of view of local algebra, we get the following more general result: