Good ideals and $${p_{g}}$$ p g -ideals in two-dimensional normal singularities

Okuma, Tomohiro; Watanabe, Kei-ichi; Yoshida, Ken-ichi

doi:10.1007/s00229-016-0821-7

Cited by 16 publications

(51 citation statements)

References 18 publications

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“…In what follows, let A, X, I = I Z be as above. The authors have studied p g -ideals in [15,16,17]. So we first recall the notion of p g -ideals in terms of q(kI).…”

Section: Normal Reduction Numbers and Geometric Genusmentioning

confidence: 99%

Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

2019

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In this paper, we give a formula for normal reduction number of an integrally closed m-primary ideal of a 2-dimensional normal local ring (A, m) in terms of the geometric genus p g (A) of A. Also we compute the normal reduction number of the maximal ideal of Brieskorn hypersurfaces. As an application, we give a short proof of a classification of Brieskorn hypersurfaces having elliptic singularities.

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“…In what follows, let A, X, I = I Z be as above. The authors have studied p g -ideals in [15,16,17]. So we first recall the notion of p g -ideals in terms of q(kI).…”

Section: Normal Reduction Numbers and Geometric Genusmentioning

confidence: 99%

Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

2019

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“…Okuma and the last two authors [13] proved that dim k H 0 (O X (−Z)) ≤ p g (A) holds true if H 0 (O X (−Z)) has no fixed component. Definition 2.10 (See [13]). An anti-nef cycle Z is a p g -cycle if O X (−Z) is generated and dim k H 0 (O X (−Z)) = p g (A).…”

Section: P G -Idealsmentioning

confidence: 99%

The strong Rees property of powers of the maximal ideal and Takahashi–Dao's question

2021

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Dedicated to Craig Huneke on the occasion of his 65th Birthday.Abstract. In this paper, we introduce the notion of the strong Rees property (SRP) for m-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2. As its application, we characterize two-dimensional excellent normal local domains so that m is a pg-ideal.Finally we ask what m-primary ideals have SRP and state a conjecture which characterizes the case when m n are the only ideals which have SRP.Question 1.2. Assume that (A, m) is a two-dimensional excellent normal local domain. If an inequality (µ(I) − 1) · ℓℓ(I) ≥ e(I) holds for any m-primary integrally closed ideal I, then is m a p g -ideal?It turns out that this question is related to the "strong Rees property" of powers of the maximal ideal. The main result in this paper is the following theorem. Theorem 3.2. Let (A, m) be a Noetherian local ring. Assume that depth A ≥ 2 and H 1 M (G) has finite length, where G = G(m) = ⊕ n≥0 m n /m n+1 . If m ℓ is Ratliff-Rush closed, then m ℓ has the strong Rees property.By Remark 2.5, any power m ℓ is Ratliff-Rush closed whenever depth G ≥ 1. Hence we have the following corollary.

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“…Thus our condition (c) says that every m-full m-primary ideal has minimal mixed multiplicity. When R is an excellent normal local domain over an algebraically closed field, then R ∈ C precisely when m is a p g -ideal in the sense of [20]. where k[[x, y, z]]/(f (x, y, z)) is a Du Val surface (i.e., a rational double point) and g is arbitrary.…”

Section: Dimensionmentioning

confidence: 99%

“…Acknowledgments. We are deeply grateful to Kei-ichi Watanabe for many detailed discussions on [20,21]. His insights gave us the geometric proofs in Section 5 and suggested the if and only if…”

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confidence: 96%

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The multiplicity and the number of generators of an integrally closed ideal

Dao¹,

Smirnov²

2019

jsing

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Let (R, m) be a Noetherian local ring and I a m-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if Spec R has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. In addition, we can classify when the inequality always hold for a Cohen-Macaulay R of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on pg ideals by Okuma-Watanabe-Yoshida, a conjecture of Huneke, Mustaţǎ, Takagi, and Watanabe on F -threshold, multiplicity of the fiber cone, and the h-vector of the associated graded ring. statement in the dimension two case of Theorem 1.1. We are also grateful to

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Good ideals and $${p_{g}}$$ p g -ideals in two-dimensional normal singularities

Cited by 16 publications

References 18 publications

Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

The strong Rees property of powers of the maximal ideal and Takahashi–Dao's question

The multiplicity and the number of generators of an integrally closed ideal

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