Geometry & Topology, Vol. 9 (2005) Paper no. 17, pages 699--755.

Complete intersection singularities of splice type as universal abelian covers

Walter D Neumann, Jonathan Wahl

Abstract. 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.

Keywords. Surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover

AMS subject classification. Primary: 32S50, 14B05. Secondary: 57M25, 57N10.

DOI: 10.2140/gt.2005.9.699

E-print: arXiv:math.AG/0407287

Submitted to GT on 31 October 2004. (Revised 18 April 2005.) Paper accepted 6 March 2005. Paper published 28 April 2005.

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

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