Complete intersection singularities of splice type as universal abelian covers

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

Mathematical Subject Classification 2000

Primary: 32S50, 14B05

Secondary: 57M25, 57N10

References

Forward citations

Publication

Received: 31 October 2004

Revised: 18 April 2005

Accepted: 6 March 2005

Published: 28 April 2005

Proposed: Robion Kirby

Seconded: Ronald Fintushel, Ronald Stern

Authors

Walter D Neumann
Department of Mathematics Barnard College Columbia University New York New York 10027 USA

Jonathan Wahl
Department of Mathematics The University of North Carolina Chapel Hill North Carolina 27599-3250 USA

Volume 9, issue 2 (2005)

Walter D Neumann and Jonathan Wahl

Abstract