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