The End Curve Theorem for normal complex surface singularities

TL;DR

The paper proves the End Curve Theorem, characterizing normal surface singularities with rational homology sphere links as splice-quotients based on the existence of end curve functions, with implications for known classes.

Contribution

It establishes a new characterization of splice-quotient singularities via end curve functions, providing explicit equations and descriptions of their universal abelian covers.

Findings

Characterization of splice-quotient singularities using end curve functions

Explicit equations for universal abelian covers of these singularities

Connections to weighted homogeneous, rational, and minimally elliptic 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).