The end curve theorem for normal complex surface singularities

Neumann, 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 -