Combinatorics of rational singularities

Tráng, Lê Dũng; Tosun, Meral

doi:10.1007/s00014-004-0808-y

Cited by 14 publications

(5 citation statements)

References 15 publications

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“…When S ⊆ C, f S is crepant and, furthermore, X S has only isolated ADE singularities since we have contracted only (−2)-curves. It is well known that in the dual graph of the minimal resolution, all maximal (−2)-curves must lie in ADE configurations (see, for example, [TT04,Proposition 3.2]).…”

Section: Tilting Bundles On Partial Resolutionsmentioning

confidence: 99%

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Frobenius categories, Gorenstein algebras and rational surface singularities

Kalck¹,

Iyama²,

Wemyss³

et al. 2014

Compositio Math.

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We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many IwanagaGorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.

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Section: Tilting Bundles On Partial Resolutionsmentioning

confidence: 99%

“…Remark 4.23. We remark that the method in the above proof cannot be applied to E 8 , since it is well known that the rational tree E 8 with all vertices labelled with −2 cannot be a (strict) subtree of any rational tree [TT04,Corollary 3.11].…”

mentioning

confidence: 99%

Frobenius categories, Gorenstein algebras and rational surface singularities

Kalck¹,

Iyama²,

Wemyss³

et al. 2014

Compositio Math.

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show abstract

“…When S ⊆ C then f S is crepant and further X S has only isolated ADE singularities since we have contracted only (−2)-curves -it is well-known that in the dual graph of the minimal resolution, all maximal (−2)-curves must lie in ADE configurations (see e.g. [TT,3.2]).…”

Section: Relationship To Partial Resolutions Of Rational Surface Sing...mentioning

confidence: 99%

“…Remark 4.23. We remark that the method in the above proof cannot be applied to E 8 , since it is well-known that the rational tree E 8 (labelled with −2's) cannot be a (strict) subtree of any rational tree [TT,3.11].…”

Section: E4mentioning

confidence: 99%

Frobenius categories, Gorenstein algebras and rational surface singularities

Iyama,

Kalck,

Wemyss

et al. 2012

Preprint

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We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.

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“…He proved simpleness for some special classes of singularities, namely, rational quadruple points or sandwiched singularities in [5]. For the classification of certain classes of rational singularities, the interested readers can refer to the recent papers [6]- [9]. In [10], Laufer examined a class of elliptic singularities that satisfy a minimality condition.…”

Section: § 1 Introductionmentioning

confidence: 99%

Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve

Yau,

Zhu,

Zuo

2023

Izv. Math.

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Let $(V, p)$ be a normal surface singularity. Let $\pi\colon (M, A)\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.

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Combinatorics of rational singularities

Cited by 14 publications

References 15 publications

Frobenius categories, Gorenstein algebras and rational surface singularities

Frobenius categories, Gorenstein algebras and rational surface singularities

Frobenius categories, Gorenstein algebras and rational surface singularities

Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve

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