The end curve theorem for normal complex surface singularities

Abstract

We prove the “End Curve Theorem,” which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\Sigma$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.

An “end curve function” is an analytic function $(X,o)\to (C,0)$ whose zero set intersects $\Sigma$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.

A “splice quotient singularity” $(X,o)$ is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in $C^t$ , where $t$ is the number of leaves in the resolution graph for $(X,o)$, together with an explicit description of the covering transformation group.

Among the immediate consequences of the End Curve Theorem are the previously known results: $(X,o)$ is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).