Generic projections, the equations defining projective varieties and Castelnuovo regularity

For a reduced, irreducible projective variety X of degree d and codimension e in the Castelnuovo-Mumford regularity is defined as the least k such that X is k-regular, i.e., for , where is the sheaf of ideals of X. There is a long standing conjecture about k-regularity (see 5]): . Here we show that for any smooth fivefold and for any smooth sixfold by extending methods used in 10]. Furthermore, we give a bound for the regularity of a reduced, connected and equidimensional locally Cohen-Macaulay curve or surface in terms of degree d, codimension e and an arithmetic genus (see Theorem 4.1). Received November 12, 1998; in final form May 4, 1999