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A CUSP SINGULARITY WITH NO GALOIS COVER BY A COMPLETE INTERSECTION
 

Summary: A CUSP SINGULARITY WITH NO GALOIS COVER BY A
COMPLETE INTERSECTION
DAVID E. ANDERSON
Abstract. With an explicit example, we confirm a conjecture by Neumann
and Wahl that there exist cusps with no Galois cover by a complete intersec-
tion. Some computational techniques are reviewed, and a method for deciding
whether a given cusp has a complete intersection Galois cover is developed.
1. Introduction
In [7], Neumann and Wahl prove that the universal abelian cover of every
quotient-cusp is a complete intersection, and conjecture generally that a similar
result holds for any Q-Gorenstein normal surface singularity whose link is a ratio-
nal homology sphere. (For discussions of these singularities and others known to
have complete intersection abelian covers, see [5, 6, 7]. By a cover of a complex
surface singularity we mean a finite cover of a germ of the singularity, branched
only at the singular point.) In the same article (Proposition 4.1), it is proved that
every cusp has a cover by a complete intersection, but there exist cusps with no
abelian cover by a complete intersection. The authors go on to conjecture that
some cusps do not even have a Galois cover by a complete intersection.
In the present paper, we exhibit a cusp confirming this latter conjecture: the
cusp classified by the cycle (8, 2, 4, 3, 12) satisfies as an example. The existence of

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics