Cohen-Macaulay coordinate rings of blowup schemes (Q1381289)

Summary: 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 \(c\geq ef\). 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.