Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

Details Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

2004-05-12

arxiv.org/abs/math/0405214v1

Mathematics

Commutative Algebra

21 pages, to appear in Trans. Amer. Math. Soc

Scientific paper

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 \subset R. It is known that there are embeddings Y \cong Proj k[(I^e)_c] for c \ge 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^0(Y,O_Y) = k, H^i(Y,O_Y) = 0 for i = 1,...,dim Y-1, Y 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 a and b such that k[(I^e)_c] is Cohen-Macaulay for all c > d(I)e + a and e > b. 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)_ct] (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 f_*O_Y = O_X, R^i f*O_Y = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(I^e)_ct] for all c > d(I)e + a and e > b. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras.

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