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September, 2003 A new Proof of Desingularization over fields of characteristic zero
Santiago Encinas, Orlando Villamayor
Rev. Mat. Iberoamericana 19(2): 339-353 (September, 2003).


We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness (see also \cite{EncinasVillamayor2000} page 224). Given a subscheme defined by equations, we prove that embedded desingularization can be achieved by a sequence of monoidal transformations; where the law of transformation on the equations defining the subscheme is simpler then that used in Hironaka's procedure. This is done by showing that desingularization of a closed subscheme $X$, in a smooth sheme $W$, is achieved by taking an algorithmic principalization for the ideal $I(X)$, associated to the embedded scheme $X$. This provides a conceptual simplification of the original proof of Hironaka. This algorithm of principalization (of Log-resolution of ideals), and this new procedure of embedded desingularization discussed here, have been implemented in MAPLE.


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Santiago Encinas. Orlando Villamayor. "A new Proof of Desingularization over fields of characteristic zero." Rev. Mat. Iberoamericana 19 (2) 339 - 353, September, 2003.


Published: September, 2003
First available in Project Euclid: 8 September 2003

zbMATH: 1073.14021
MathSciNet: MR2023188

Primary: 14E15 , 32S45

Keywords: desingularization , resolution of singularities

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.19 • No. 2 • September, 2003
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