# One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Open Mathematics (2011)

• Volume: 9, Issue: 6, page 1349-1353
• ISSN: 2391-5455

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## Abstract

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Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.

## How to cite

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Veronique Lierde. "One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal." Open Mathematics 9.6 (2011): 1349-1353. <http://eudml.org/doc/268961>.

@article{VeroniqueLierde2011,
abstract = {Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.},
author = {Veronique Lierde},
journal = {Open Mathematics},
keywords = {Degree function; Rees valuation; 2-dimensional rational singularity},
language = {eng},
number = {6},
pages = {1349-1353},
title = {One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal},
url = {http://eudml.org/doc/268961},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Veronique Lierde
TI - One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1349
EP - 1353
AB - Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.
LA - eng
KW - Degree function; Rees valuation; 2-dimensional rational singularity
UR - http://eudml.org/doc/268961
ER -

## References

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