The end curve theorem for normal complex surface singularities

@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 -