On the structure of local cohomology modules for monomial curves in

Abstract

Our setting for this paper is projective 3-space over an algebraically closed field K . By a curve C ⊂ is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Let be the Hartshorne-Rao module of finite length (cf. [R]). Here Z is the set of integers and ℐ c the ideal sheaf of C . In [GMV] it is shown that , where is the homogeneous ideal of C , is the first local cohomology module of the R -module M with respect to . Thus there exists a smallest nonnegative integer k ∊ N such that , (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown that k = 0 if and only if C is arithmetically Cohen-Macaulay and C is arithmetically Buchsbaum if and only if k ≤ 1. We therefore have the following natural definition.