Symmetries of Surface Singularities

Müller, Gerd

doi:10.1112/s0024610799007188

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…If f belongs to a reductive algebraic subgroup G of Aut(Y ), then our result would follow since the Zariski closure of f in G is a subgroup isomorphic to C * acting effectively on Y , and a theorem of Scheja and Wiebe [SW81, Satz 3.1] implies Y to be a weighted homogeneous singularity, see [Mül99,Theorem 1]. If f is the time-one map of the flow of a holomorphic vector field having a dicritical singularity at 0, our theorem is then due to Camacho, Movasati and Scardua [CMS09].…”

Section: Introductionmentioning

confidence: 92%

Normal surface singularities admitting contracting automorphisms

Favre¹,

Ruggiero²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: Introductionmentioning

confidence: 92%

Normal surface singularities admitting contracting automorphisms

Favre¹,

Ruggiero²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…The classification statements regarding topological types of normal surface singularities are important for some potential applications in our future work. In fact, these classes of singularities are the most interesting ones in the study of symmetries of surface singularities [21], since there is a natural homomorphism from the automorphism group of the singularities to the automorphism group of the central curve (that is, the −3-curve). Furthermore, we compute the fundamental cycles of maximal graphs.…”

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

Yau,

Zhu,

Zuo

2023

Известия Российской академии наук. Серия математическая

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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 to $A$ 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 $-3$-curve, and the rest are all $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see Section 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give the formulas for computing arithmetic and geometric genera of star-shaped graphs. Bibliography: 28 titles.

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Symmetries of Surface Singularities

Cited by 3 publications

References 31 publications

Normal surface singularities admitting contracting automorphisms

Normal surface singularities admitting contracting automorphisms

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

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

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