A new proof of desingularization over fields of characteristic zero.

By Santiago Encinas, Orlando Villamayor. 2001

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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.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f9a6bd631124cf2cec05cd04206a9764054a8233

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Citations in the corpus

A Simplified Proof of Desingularization and Applications (2002, methodology): “The purpose of this paper is threefold: We present a simplified proof of both theorems, already indicated in [18], see also [19] (this is done in Parts II and V of this manuscript); we summarize some applications of our approach, together with new results (see Part III); and finally, in Part IV, we ”
Resolution of singularities for 3-folds in positive characteristic (2006, methodology): “Several simplifications of Hironaka’s proof of characteristic zero resolution have appeared, making the proof quite accessible now, including Abramovich and de Jong [AJo], Bierstone and Milman [BrM], Bogomolov and Pantev [BP], Bravo, Encinas and Villamayor [BEV], Encinas and Hauser [EH], Encinas and”