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 $Σ$ 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 $Σ$ 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).