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

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

doi:10.1007/s40306-018-00311-4

Cited by 7 publications

(10 citation statements)

References 14 publications

(24 reference statements)

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“…This makes the bridge between the theory of the normal Hilbert coefficients and the theory of the singularities. This is the line already traced by Lipman [16], Cutkosky [4], and more recently by Okuma, Watanabe, and Yoshida (see [23]–[25]).…”

Section: Introduction and Notationssupporting

confidence: 68%

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Normal Hilbert Coefficients and Elliptic Ideals in Normal Two-Dimensional Singularities

Okuma¹,

Rossi²,

Watanabe³

et al. 2022

Nagoya Math. J.

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Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

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Section: Introduction and Notationssupporting

confidence: 68%

“…Definition 2.4 (cf. [26]). Let I ⊂ A be m-primary integrally closed ideal, and let Q be a minimal reduction of I.…”

Section: §1 Introduction and Notationsmentioning

confidence: 99%

Normal Hilbert Coefficients and Elliptic Ideals in Normal Two-Dimensional Singularities

Okuma¹,

Rossi²,

Watanabe³

et al. 2022

Nagoya Math. J.

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“…For any minimal reduction Q of I, we have that I n+1 = QI n for all large n. We define the normal reduction number r(I) of I by r(I) := min{r | I n+1 = QI n for all n ≥ r and a minimal reduction Q}; this integer does not depend on the choice of Q (cf. [H87,OWY19a]). The normal reduction number of (X, o) is defined by r(X, o) := max{r(I)…”

Section: Introductionmentioning

confidence: 99%

“…However, the structure of the set of reductions is clarified only for very special cases (see e.g. [OWY19a,OWY19b,O19]).…”

Section: Introductionmentioning

confidence: 99%

Normal reduction number of normal surface singularities

Nagy,

Némethi,

Okuma

2021

Preprint

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Let (X, o) be a complex analytic normal surface singularity and let O X,o be its local ring. We investigate the normal reduction number of O X,o and related numerical analytical invariants via resolutions X → X of (X, o) and cohomology groups of different line bundles L ∈ Pic( X).The normal reduction number is the universal optimal bound from which powers of certain ideals have stabilization properties. Here we combine this with stability properties of the iterated Abel maps. Some of the main results provide topological upper bounds for both stabilization properties.The present note was partially motivated by the open problems formulated in [O19]. Here we answer several of them.

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“…This study was carried on by J. Lipman, B. Teissier and more recently by T. Okuma, K.-i. Watanabe, and K.Yoshida with the study of p g -ideals which inherit nice properties of integrally closed ideals on rational singularities, see [16,17].…”

Section: Introductionmentioning

confidence: 99%

On the structure of the Sally module and the second normal Hilbert coefficient

Masuti

Ozeki

Rossi

et al. 2020

Proc. Amer. Math. Soc.

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The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third Hilbert coefficient has been necessary. This study is related to a long-standing conjecture stated by S. Itoh. 2010 Mathematics Subject Classification. 13D40, 13A30, 13H10. Proposition 2.4. Let d ≥ 2. Then the graded T -module C (2) J ({I n } n∈Z ) satisfies the Serre's property (S 2 ) as a T /Ann T (C (2) J ({I n } n∈Z ))-module. 3. The structure of the Sally module when e 3 (I) = 0In this section we prove the first main result of this paper (Theorem 3.6). We set C = C J (I) = C(2) J ({I n } n∈Z ). In the following theorem we recall few results on the vanishing of local cohomology modules from [11] (see also [6]). From now onwards we set M ′ = (t −1 , m, It)R ′ (I) for the graded maximal ideal of R ′ (I) and N ′ = ItR ′ (I).

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

Cited by 7 publications

References 14 publications

Normal Hilbert Coefficients and Elliptic Ideals in Normal Two-Dimensional Singularities

Normal Hilbert Coefficients and Elliptic Ideals in Normal Two-Dimensional Singularities

Normal reduction number of normal surface singularities

On the structure of the Sally module and the second normal Hilbert coefficient

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