Generic projections, the equations defining projective varieties and Castelnuovo regularity

Abstract

For a reduced, irreducible projective variety X of degree d and codimension c in P-N the Castelnuovo-Mumford regularity regX is defined as the least k such that X is k-regular, i.e., H-i(P-N, I-X(k - i)) = 0 for i greater than or equal to 1, where I-X subset of O-PN is the sheaf of ideals of X. There is a long standing conjecture about k-regularity (see [5]): regX less than or equal to d - e + 1. Here we show that regX less than or equal to (d - e + 1) +10 fur any smooth fivefold and regX less than or equal to (d - e + 1) + 20 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 c and an arithmetic genus rho(a) (see Theorem 4.1).