Cohen-Macaulay coordinate rings of blowup schemes

Abstract

Suppose that Y is a projective k-scheme with Cohen-Macaulay coordinate ring S. Let $I\subset S$ be a homogeneous ideal of S. I can be blown up to produce a projective k-scheme X which birationally dominates Y. Let I _c be the degree c part of I. Then k[I _c ] is a coordinate ring of a projective embedding of X for all c sufficiently large. This paper considers the question of when there exists a constant f such that k[(I ^e) _c ] is Cohen-Macaulay for $c\ge ef$. 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.