Filtered Rings, Filtered Blowing-Ups and Normal Two-Dimensional Singularities with “Star-Shaped” Resolution

In this paper, we study local rings of small Hilbert-Kunz multiplicity. In particular, we prove that an unmixed local ring of Hilbert-Kunz multiplicity one is regular and classify two-dimensional Cohen-Macaulay local rings whose Hilbert-Kunz multiplicity is 2 or less. Also, we investigate the inequality between the multiplicity and the colength of the tight closure of parameter ideals inverse to the usual inequality between multiplicity and colength.

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“…Now, consider the filtration F l = H 0 X −lE 0 on A. Then it is shown in [TW1,6.3] that the associated graded ring…”

Section: 3mentioning

confidence: 99%

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Watanabe

Yoshida

2000

Journal of Algebra

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“…The proof of (2)-(4) is same as the proof of [20, (3.3-3.4)] based on part (1) and filtered ring theory [24].…”

Section: Filtration With Respect To Ev the V-order Defines A Filtratmentioning

confidence: 86%

On the Casson Invariant Conjecture of Neumann–Wahl

Némethi

Okuma

2008

J. Algebraic Geom.

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Abstract. In the article we prove the Casson Invariant Conjecture of NeumannWahl for splice type surface singularities. Namely, for such an isolated complete intersection, whose link is an integral homology sphere, we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber. IntroductionAlmost twenty years ago, Neumann and Wahl formulated the following conjecture: an isolated complete intersection surface singularity whose link Σ is an integral homology 3-sphere Then the Casson invariant λ(Σ) of the link is one-eighth the signature of the Milnor fiber of (X, o).The conjecture can be reformulated in terms of the geometric genus pg of (X, o) as well (see below in 4.1).The conjecture is true for Brieskorn hypersurface singularities by a result of Fintushel and Stern [8]. This and additivity properties (with respect to splice decomposition) lead to the verification of the conjecture for Brieskorn complete intersections, done independently by Neumann-Wahl [15] and Fukuhara-Matsumoto-Sakamoto [9]. For suspension hypersurface singularities it was verified in [15]. Some iterative generalizations, related with cyclic coverings and using techniques of equivariant Casson invariant and gauge theory, were covered by Collin and Saveliev (cf. [3,4,5]).Recently, Neumann and Wahl have introduced an important family of complete intersection surface singularities, the splice type singularities. In [18] they treated the case when the link is an integral homology sphere (the reader may consult in [16,17,19] the case of rational homology sphere links too). In [18], they have also verified the above conjecture (by a direct computation of the geometric genus) for special splice type singularities (when the nodes of the splice diagram 'are in a line').The goal of the present article is to verify the conjecture for an arbitrary splice type singularity:Theorem. The Casson Invariant Conjecture is true for any splice type singularity with integral homology sphere link.

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“…Now let us compute the fundamental cycle Z on an exceptional set A with star-shaped dual graph and the fundamental genus pf. Let m be the Further, from Lemma 6.14 and (6.15) in [15], we can see the followings:…”

Section: Fundamental Genus Of Normal Surface Singularities Withmentioning

confidence: 99%

On Gorenstein surface singularities with fundamental genusp_f≧ 2 which satisfy some minimality conditions

Tomaru¹

1995

Pacific J. Math.

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Filtered Rings, Filtered Blowing-Ups and Normal Two-Dimensional Singularities with “Star-Shaped” Resolution

Cited by 39 publications

References 33 publications

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

On the Casson Invariant Conjecture of Neumann–Wahl

On Gorenstein surface singularities with fundamental genusp_f≧ 2 which satisfy some minimality conditions

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Filtered Rings, Filtered Blowing-Ups and Normal Two-Dimensional Singularities with “Star-Shaped” Resolution

Cited by 39 publications

References 33 publications

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

On the Casson Invariant Conjecture of Neumann–Wahl

On Gorenstein surface singularities with fundamental genuspf≧ 2 which satisfy some minimality conditions

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Product

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On Gorenstein surface singularities with fundamental genusp_f≧ 2 which satisfy some minimality conditions