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Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

2004/05/12 by Huy Tài Hà, Huy Tai Ha, Ha, Huy Tai +3
Mathematics · #13A30 #13H10 #14M05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13A30 #msc:13H10 #msc:14M05

paper · pdf · doi:10.48550/arxiv.math/0405214

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

arxiv created 2004/05/12 · openalex publication_date 2004/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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[(Ie)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[(Ie)c] if and only if H0(Y,OY) = k, Hi(Y,OY) = 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[(Ie)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[(Ie)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[(Ie)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_*OY = OX, Ri f*OY = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(Ie)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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