# The end curve theorem for normal complex surface singularities

Walter D. Neumann; Jonathan Wahl

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 2, page 471-503
- ISSN: 1435-9855

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topNeumann, Walter D., and Wahl, Jonathan. "The end curve theorem for normal complex surface singularities." Journal of the European Mathematical Society 012.2 (2010): 471-503. <http://eudml.org/doc/277377>.

@article{Neumann2010,

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 $(X,o)\rightarrow (\mathbb \{C\},0)$ 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 $\mathbb \{C\}^t$, 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).},

author = {Neumann, Walter D., Wahl, Jonathan},

journal = {Journal of the European Mathematical Society},

keywords = {surface singularity; splice quotient singularity; rational homology sphere; complete intersection singularity; abelian cover; numerical semigroup; monomial curve; linking pairing; surface singularity; splice quotient singularity; rational homology sphere; complete intersection singularity; abelian cover; numerical semigroup; monomial curve; linking pairing},

language = {eng},

number = {2},

pages = {471-503},

publisher = {European Mathematical Society Publishing House},

title = {The end curve theorem for normal complex surface singularities},

url = {http://eudml.org/doc/277377},

volume = {012},

year = {2010},

}

TY - JOUR

AU - Neumann, Walter D.

AU - Wahl, Jonathan

TI - The end curve theorem for normal complex surface singularities

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 2

SP - 471

EP - 503

AB - 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 $(X,o)\rightarrow (\mathbb {C},0)$ 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 $\mathbb {C}^t$, 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).

LA - eng

KW - surface singularity; splice quotient singularity; rational homology sphere; complete intersection singularity; abelian cover; numerical semigroup; monomial curve; linking pairing; surface singularity; splice quotient singularity; rational homology sphere; complete intersection singularity; abelian cover; numerical semigroup; monomial curve; linking pairing

UR - http://eudml.org/doc/277377

ER -

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