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Complete intersection singularities of splice type as universal abelian covers

Walter D Neumann and Jonathan Wahl

Geometry & Topology 9 (2005) 699–755

DOI: 10.2140/gt.2005.9.699

arXiv: math.AG/0407287


It has long been known that every quasi-homogeneous normal complex surface singularity with Q–homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q–homology sphere links, called “splice-quotient singularities.” According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with Q–homology sphere links. As quotients of complete intersections, they are necessarily Q–Gorenstein, and many Q–Gorenstein singularities with Q–homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q–homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.


surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover

Mathematical Subject Classification

Primary: 14B05, 32S50

Secondary: 57M25, 57N10

Forward citations

Received: 31 October 2004
Revised: 18 April 2005
Accepted: 6 March 2005
Published: 28 April 2005
Proposed: Robion Kirby
Seconded: Ronald Fintushel, Ronald Stern

Walter D Neumann
Department of Mathematics
Barnard College
Columbia University
New York
New York 10027
Jonathan Wahl
Department of Mathematics
The University of North Carolina
Chapel Hill
North Carolina 27599-3250