An approach to the uniform Artin-Rees theorems from the notion of relation type

Planas-Vilanova, Francesc

doi:10.1201/9781420028324.ch13

Cited by 2 publications

(4 citation statements)

References 31 publications

(56 reference statements)

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“…Let X be a set of 6 points in P 2 k . Then the Jacobian ideal of I(X) is not V V-torsion-free if and only if X is one of the configurations (2),( 3),( 7), (11).…”

Section: Letmentioning

confidence: 99%

“…The Aluffi algebra is an algebraic version of characteristic cycles in intersection theory [6]. The connection of module of Valabrega-Valla with the Aluffi algebra is as follows: The vanishing of V V J⊂I has close relation with the theory of I-standard base ( in the sense of Hironaka), Artin Rees number and relation type number (See [11] and [10]).…”

Section: Introductionmentioning

confidence: 99%

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The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Nejad,

Shahidi

2019

Preprint

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The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety V is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where V is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -called (s − r)-fold collinear configuration. A complete classification of types is given for 5 and 6 points in regard to this problem.

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“…Let X be a set of 6 points in P 2 k . Then the Jacobian ideal of I(X) is not V V-torsion-free if and only if X is one of the configurations (2),( 3),( 7), (11).…”

Section: Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Nejad,

Shahidi

2019

Preprint

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“…There are several cases in which the strong uniform Artin-Rees property holds ( [12,3,13,9,15]), but Wang has shown in [22] that it does not hold in general. See [14] for a recent summary and explication of results relating to uniform Artin-Rees theorems.…”

Section: Introductionmentioning

confidence: 99%

“…See [Planas-Vilanova 2006] for a recent summary and explication of results relating to uniform Artin-Rees theorems.…”

Section: Introductionmentioning

confidence: 99%

Homology multipliers and the relation type of parameter ideals

Aberbach¹,

Ghezzi²,

Harbourne³

2006

Pacific J. Math.

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The relation type question, raised by C. Huneke, asks whether for a complete equidimensional local ring R there exists a uniform number N such that the relation type of every ideal I ⊂ R generated by a system of parameters is at most N. Wang gave a positive answer to this question when the non-Cohen-Macaulay locus of R (denoted by NCM(R)) has dimension zero. In this paper, we first present an example, due to the first author, which gives a negative answer to the question when dim NCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/Ꮽ(R) is a domain, where Ꮽ( R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.

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An approach to the uniform Artin-Rees theorems from the notion of relation type

Cited by 2 publications

References 31 publications

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Homology multipliers and the relation type of parameter ideals

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