Universal abelian covers of certain surface singularities

2005/03/31 by Okuma, Tomohiro
#14J17 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 32S25 #Secondary 14B05

paper · doi:10.48550/arxiv.math/0503733

Abstract

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 (Y₀,o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G.

Related

• Universal abelian covers of surface singularities
• Universal abelian covers of superisolated singularities
• Topology, geometry, and equations of normal surface singularities
• Splice Diagram Singularities and The Universal Abelian Cover of Graph Orbifolds
• The multiplicity of abelian covers of splice quotient singularities