Cohomology of ideals in elliptic surface singularities

Okuma, Tomohiro

doi:10.1215/ijm/1534924827

Cited by 8 publications

(12 citation statements)

References 23 publications

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“…Remark 4.1.3. Using the relevant exact sequences one verifies that n 0 (Z, L) = min{n ≥ 0 : [O17,Lemma 3.6].…”

Section: Stability Bound Formentioning

confidence: 99%

“…One has the following relations between the 'stabilized dimensions'. By [O17,Remark 3.8] or [NNI,Th. 6.1.9] we know that for L without fixed components lim n→∞ h 1 (Z,…”

Section: Stability Bound Formentioning

confidence: 99%

“…In fact, n 0 (Z, L) = min{n ≥ 0 : h 1 (Z, L n ) = h 1 (Z, L m ), ∀m > n} as well, see e.g. [O17,Lemma 3.6].…”

Section: Introductionmentioning

confidence: 99%

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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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“…Remark 4.1.3. Using the relevant exact sequences one verifies that n 0 (Z, L) = min{n ≥ 0 : [O17,Lemma 3.6].…”

Section: Stability Bound Formentioning

confidence: 99%

“…One has the following relations between the 'stabilized dimensions'. By [O17,Remark 3.8] or [NNI,Th. 6.1.9] we know that for L without fixed components lim n→∞ h 1 (Z,…”

Section: Stability Bound Formentioning

confidence: 99%

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Normal reduction number of normal surface singularities

Nagy,

Némethi,

Okuma

2021

Preprint

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“…Okuma proved in [Ok,Theorem 3.3] that if A is an elliptic singularity, then r(A). For the definition of elliptic singularity we refer to [W, page 428] or [Ok,Definition 2.1].…”

Section: Elliptic and Strongly Elliptic Idealsmentioning

confidence: 99%

“…Okuma proved in [Ok,Theorem 3.3] that if A is an elliptic singularity, then r(A). For the definition of elliptic singularity we refer to [W, page 428] or [Ok,Definition 2.1]. We investigate the integrally closed m-primary ideals such that r(I) = 2 with the aim to characterize elliptic singularities.…”

Section: Elliptic and Strongly Elliptic Idealsmentioning

confidence: 99%

Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities

Okuma¹,

Rossi²,

Yoshida³

2020

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Let (A, m) be an excellent two-dimensional normal local domain. The geometric genus pg(A) is an important geometric invariant of A. A rational singularity is characterized by pg(A) = 0 and the integrally closed m-primary ideals of A are normal and well described by Cutkosky and Lipman. Later, Okuma, Watanabe and Yoshida characterized rational singularities through the pg-ideals. In this paper we define 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 by Yau. A strongly elliptic singularity can be described by pg(A) = 1. We characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed m-primary ideals of A. Unlike pg-ideals, elliptic and strongly elliptic ideals are not necessarily normal and we give necessary and sufficient conditions for being normal. In the last section we discuss the existence (and the effective construction) of strongly elliptic ideals in any 2-dimensional normal local ring.

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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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Cohomology of ideals in elliptic surface singularities

Cited by 8 publications

References 23 publications

Normal reduction number of normal surface singularities

Normal reduction number of normal surface singularities

Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities

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

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