M ay 2 00 4 ASYMPTOTIC BEHAVIOUR OF ARITHMETICALLY COHEN - MACAULAY BLOW - UPS

This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I ⊂ R. It is known that there are embeddings Y ∼ = Proj k[(I e) c ] for c ≥ d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I e) c ] if and only if H is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k[(I e) c ] is a Cohen-Macaulay ring. We obtain a surprising compelte answer to this question, namely, that under the above conditions, there are well determined invariants ε and e 0 such that k[(I e) c ] is Cohen-Macaulay for all c > d(I)e + ε and e > e 0. Our approach is based on recent results on the asymptotic linearity of the Castelnuovo-Mumford regularity of ideal powers. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I e) c t] (which provides an arithmetic Macaulayfication for X). If R has negative a *-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if π * O Y = O X , R i π * O Y = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(I e) c t] for all c > d(I)e + ε and e > e 0. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras.