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| Format: | Preprint |
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2005
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| Online Access: | https://arxiv.org/abs/math/0503733 |
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| _version_ | 1866917147527086080 |
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| author | Okuma, Tomohiro |
| author_facet | Okuma, Tomohiro |
| contents | Every normal complex surface singularity with $\mathbb Q$-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following.
If $(X,o)$ is a rational or minimally elliptic singularity, then its universal abelian cover $(Y,o)$ is an equisingular deformation of an isolated complete intersection singularity $(Y_0,o)$ defined by a Neumann-Wahl system. Furthermore, if $G$ denotes the Galois group of the covering $Y \to X$, then $G$ also acts on $Y_0$ and $X$ is an equisingular deformation of the quotient $Y_0/G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0503733 |
| institution | arXiv |
| publishDate | 2005 |
| record_format | arxiv |
| spellingShingle | Universal abelian covers of certain surface singularities Okuma, Tomohiro Algebraic Geometry Primary 32S25, Secondary 14B05, 14J17 Every normal complex surface singularity with $\mathbb Q$-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If $(X,o)$ is a rational or minimally elliptic singularity, then its universal abelian cover $(Y,o)$ is an equisingular deformation of an isolated complete intersection singularity $(Y_0,o)$ defined by a Neumann-Wahl system. Furthermore, if $G$ denotes the Galois group of the covering $Y \to X$, then $G$ also acts on $Y_0$ and $X$ is an equisingular deformation of the quotient $Y_0/G$. |
| title | Universal abelian covers of certain surface singularities |
| topic | Algebraic Geometry Primary 32S25, Secondary 14B05, 14J17 |
| url | https://arxiv.org/abs/math/0503733 |