Variations on the Grothendieck–Serre Formula for Hilbert Functions and Their Applications

Masuti, Shreedevi K.; Sarkar, Parangama; Verma, J. K.

doi:10.1007/978-981-10-1651-6_8

Cited by 3 publications

(6 citation statements)

References 29 publications

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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: 69%

“…A rich literature is available on the normal Hilbert coefficients ēi (I), and this study is considered an important part of the theory of blowing-up rings (see, e.g., [2], [3], [10], [11], [12], [17]- [19], [33]).…”

Section: §1 Introduction and Notationsmentioning

confidence: 99%

“…The normal reduction number exists (see [21], [29]), and it has been studied by many authors in the context of the Hilbert function and of the Hilbert polynomial (see, e.g., [2], [3], [10], [12], [19]). The main difficulty of the normal filtration with respect to the I -adic filtration is that the Rees algebra of the normal filtration is not generated by the part of degree one because II n = I n+1 .…”

Section: §1 Introduction and Notationsmentioning

confidence: 99%

“…From the geometric side, any integrally closed -primary ideal I of is represented on some resolution (see [16]). Let be a resolution of singularities with an anti-nef cycle on X , so that and .…”

Section: Introduction and Notationsmentioning

confidence: 99%

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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: 69%

Section: §1 Introduction and Notationsmentioning

confidence: 99%

Section: §1 Introduction and Notationsmentioning

confidence: 99%

Section: Introduction and Notationsmentioning

confidence: 99%

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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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“…We remark that √ M = Ḡ+ in Ḡ, hence by [MSV,Proposition 3.1. ] we have [H 2 M ( Ḡ)] 0 ∼ = I 2 /QI ∼ = A/m has length 1.…”

Section: Elliptic and Strongly Elliptic Idealsmentioning

confidence: 97%

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

Okuma¹,

Rossi²,

Yoshida³

2020

Preprint

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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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Implicit Representations of Rational Curves and Surfaces

Busé

Catanese

Postinghel

2022

SISSA Springer Series

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Variations on the Grothendieck–Serre Formula for Hilbert Functions and Their Applications

Cited by 3 publications

References 29 publications

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

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

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

Implicit Representations of Rational Curves and Surfaces

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