# The end curve theorem for normal complex surface singularities

### Walter D. Neumann

Barnard College, Columbia University, New York, USA### Jonathan Wahl

University of North Carolina at Chapel Hill, United States

## 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*) → (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).