Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

Abstract

Abstract. This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I ⊂ R. IntroductionLet X be a projective scheme over a field k. An arithmetic Macaulayfication of X is a proper birational morphism π : Y → X such that Y has an arithmetically Cohen-Macaulay embedding, i.e. there exists a Cohen-Macaulay standard graded k-algebra A such that Y ∼ = Proj A. Inspired by the problem of resolution of singularities, it was asked when X has an arithmetic Macaulayfication. The local version of this problem (arithmetic Macaulayfication of local rings) has been extensively studied in the literature and recently solved by Kawasaki [22]. An important aspect of the global problem is to determine, given a proper birational morphism Y → X, if Y has an arithmetically Cohen-Macaulay embedding, and if it does, which embeddings of Y are arithmetically Cohen-Macaulay.Let R be a standard graded k-algebra and let I ⊂ R be a homogeneous ideal such that X = Proj R and Y is the blow-up of X along the ideal sheaf of I. It was observed by Cutkosky and Herzog [9] • Y is equidimensional and Cohen-Macaulay,In the first part of this paper, we study The invariant e 0 is a projective version of the a * -invariant, which is the largest non-vanishing degree of the graded local cohomology modules [29,32]. The invariant ε comes from the asymptotic linearity of the CastelnuovoMumford regularity of powers of ideals ([31, 10, 23, 34]). We will see that the bounds c > d(I)e + ε and e > e 0 are the best possible (Theorem 2.3 and Example 2.5). The existence of linear bounds on c and e is not hard to prove. The novelty we claim here is that an explicit description for the best possible bounds is obtained. Moreover, if the Rees algebra R[It] is locally Cohen-Macaulay on X, then e 0 = 0 and we can replace the second condition by the weaker condition thatThese results strengthen and unify all previously-known results on the Cohen-Macaulayness of k[(I e ) c ] which were obtained by different methods.In the second part of this paper, we investigate the more difficult question of when Y is an arithmetically Cohen-Macaulay blow-up of X; that is, when there exists a standard graded k-algebra R and an ideal J ⊂ R, such that X = Proj R, Y is the blow-up of X along the ideal sheaf of J, and R[Jt] is a Cohen-Macaulay ring. Given R and I, we will concentrate on ideals J ⊆ I which are generated by the elements of (I The paper is organized as follows. In Section 1, we introduce the notion of a projective a * -invariant which governs how sheaf cohomology behaves through blowing up morphisms. The material in this section is interesting on its own right. In Section 2, we study the Cohen-Macaulayness of rings of the form k[(I e ) c ] which correspond to projective embeddings of Y . The last section of the paper deals with the problem of when Y is an arithmetically Cohen-Macaulay blow-up ...

Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals

“…We shall extend these notions to define the fiber a-invariants and regularity of F over the morphism π as follows. Our construction is slightly more general than that given in [10], where similar invariants, introduced in [19,20], were generalized to coherent sheaves over P(E) for a locally free coherent sheaf E of finite rank on X. Definition 3.2.…”

“…Particularly, in Theorem 3.11, we establish basic vanishing and nonvanishing properties of sheaf cohomology groups of F that are controlled by this invariant. Such an invariant a * π (F ) was first introduced in [20] in the study of arithmetic Macaulayfication of projective schemes, and later used in [10,11,12,19] in studying the a * -invariant and regularity of powers of ideals.…”