On splice quotients of the form {zⁿ = f (x,y )}

Abstract

The splice quotients, defined by W. D. Neumann and J. Wahl, are an interesting class of normal surface singularities with rational homology sphere links. In general, it is difficult to determine whether or not a singularity is analytically isomorphic to a splice quotient, although there are certain necessary topological conditions. Let {z n = f (x, y)} define a surface X f ,n with an isolated singularity at the origin in C 3 . We show that for irreducible f , if (X f ,n , 0) satisfies the necessary topological conditions, then there exists a splice quotient of the form (Xg,n , 0), where the plane curve singularity defined by g = 0 has the same topological type as the one defined by f = 0. We also present an example of an (X f ,n , 0) that is not a splice quotient, but for which the universal abelian cover is a complete intersection of splice type together with a non-diagonal action of the discriminant group.