Universal abelian covers of certain surface singularities

Details Universal abelian covers of certain surface singularities Universal abelian covers of certain surface singularities

2005-03-31

arxiv.org/abs/math/0503733v2

Math. Ann., 334, 753 - 773 (2006)

Mathematics

Algebraic Geometry

18 pages

Scientific paper

10.1007/s00208-005-0693-8

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 $Y_0/G$.

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