# 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).

## Cite this article

Walter D. Neumann, Jonathan Wahl, The end curve theorem for normal complex surface singularities. J. Eur. Math. Soc. 12 (2010), no. 2, pp. 471–503

DOI 10.4171/JEMS/206