Cohen-Macaulay coordinate rings of blowup schemes

Suppose that Y is a projective k-scheme with Cohen{Macaulay coordinate ring S. Let I S be a homogeneous ideal of S. I can be blown up to produce a projective k-scheme X which birationally dominates Y.L et I cbe the degreec part of I.T hen k ( I c) is a coordinate ring of a projective embedding of X for all c suciently large. This paper considers the question of when there exists a constant f such that k((I e )c) is Cohen{Macaulay for c ef. A very general result is proved, giving a simple criterion for a linear bound of this type. As a consequence, local complete intersections have this property, as well as many other ideals.