On constructive desingularization

In this note we sketch the proof of desingularization over fields of characteristic zero in [Encinas, S., Villamayor, O., A new theorem of desingularization over fields of characteristic zero, Preprint: arXiv:math.AG/0101208]; and we refer to [Bravo, A., Encinas, S., Villamayor, O., A simplified proof of desingularization and applications. RevistaMatemáticaIberomericana (in press)], or [Villamayor, O., Desingularization: An introduction for the young algebraist (Notes)], for full details. This proof, defined in terms of an algorithm, provides a conceptual simplification over that of Hironaka. In fact Hironaka’s notions of standard basis, Hilbert Samuel functions, and normal flatness are avoided.