# A new Proof of Desingularization over fields of characteristic zero

### Santiago Encinas

Universidad de Valladolid, Spain### Orlando Villamayor U.

Universidad Autónoma de Madrid, Spain

## 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.

## Cite this article

Santiago Encinas, Orlando Villamayor U., A new Proof of Desingularization over fields of characteristic zero. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 339–353

DOI 10.4171/RMI/350