Irrational asymptotic behaviour of Castelnuovo-Mumford regularity

Cutkosky, Steven Dale

doi:10.1515/crll.2000.043

Cited by 82 publications

(125 citation statements)

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“…In the case of the Fermat ideals, it turns out that this is also given by a linear function for high enough powers, as we will show in Theorem 3.10. By contrast, in general it can only be shown as in [CHT,Theorem 4.3] that, if R s (I) is finitely generated, then reg(I (m) ) is a periodic linear function for m large enough, i.e. there exist integers a i and b i such that reg(I (m) ) = a i m + b i for t ≡ i mod n and t ≫ 0.…”

Section: Such Column Vectors S J Do Existmentioning

confidence: 99%

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Nagel¹,

Seceleanu²

2016

Journal of Algebra

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Abstract. Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [DST]. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo-Mumford regularity of powers of Fermat ideals and we determine their reduction ideals.

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Section: Such Column Vectors S J Do Existmentioning

confidence: 99%

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Nagel¹,

Seceleanu²

2016

Journal of Algebra

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“…On the other hand, since F is a finitely generated graded module over S = R [It], there exists a number n 0 such that F n = I n−n 0 F n 0 for n ≥ n 0 . It was recently discovered that for any finitely generated graded R-module E, the CastelnuovoMumford regularity reg(I n E) is bounded by a linear function on n with slope d(I) [34, Theorem 2.2] (see also [10,23] for the case R is a polynomial ring). By definition, we always have…”

Section: Corollary 14 Assume That Y Is Locally Arithmetic Cohen-macmentioning

confidence: 99%

“…It would be of interest to find such a bound. In general, if we happen to know the minimal free resolution of S over a bi-graded polynomial ring, then we can estimate ε(I) in terms of the shifts of syzygy modules of the resolution [10].…”

Section: 52(1) and 262]) From This It Follows That [Hmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

Harbourne

Trung

2005

Trans. Amer. Math. Soc.

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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 ...

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“…, x n ]. Cutkosky, Herzog and Trung [7] and, independently, Kodiyalam [10] proved in that the CastelnuovoMumford regularity reg(I a ) of the powers of I is a linear function of a ∈ N for large a. In [7,Remark pg.252] it is asserted that the same result holds for products of powers of ideals I 1 , .…”

Section: Introductionmentioning

confidence: 98%

A remark on regularity of powers and products of ideals

Bruns

Conca

2017

Journal of Pure and Applied Algebra

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ABSTRACT. We give a simple proof for the fact that the Castelnuovo-Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals.

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Irrational asymptotic behaviour of Castelnuovo-Mumford regularity

Cited by 82 publications

References 20 publications

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

A remark on regularity of powers and products of ideals

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