On the structure of local cohomology modules for monomial curves in

Abstract

Our setting for this paper is projective 3-spaceover an algebraically closed fieldK. By a curveC⊂is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Letbe the Hartshorne-Rao module of finite length (cf. [R]). HereZis the set of integers andℐcthe ideal sheaf ofC. In [GMV] it is shown that, whereis the homogeneous ideal ofC,is the first local cohomology module of theR-moduleMwith respect to. Thus there exists a smallest nonnegative integerk∊Nsuch that, (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown thatk= 0 if and only ifCis arithmetically Cohen-Macaulay andCis arithmetically Buchsbaum if and only ifk≤ 1. We therefore have the following natural definition.