A Simplified Proof of Desingularization and Applications
By Ana Bravo, Santiago Encinas, Orlando Villamayor. 2002
98 citations recorded by the source corpus.
This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal I(X) associated to the embedded scheme X. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/ccb2c9a33a4f1bd4ea353e1ef353f9d8d0466c5d
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In Geometry and Topology
- Steven Dale Cutkosky
- Orlando Villamayor
- Santiago Encinas
- Ana Bravo
- Herwig Hauser
- J. Włodarczyk
- Shreeram S. Abhyankar
- Chi Li
In Statistical and Nonlinear Physics
Citations in the corpus
- 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 a”
- Introduction to Shimura Varieties (2004, unknown): “According Hironaka’s (Fields medal) theorem on the resolution of singularities (Hironaka 1964; see also Bravo et al. 2002), we can realize V as an open subvariety of a projective nonsingular variety V ∗ in such a way that V ∗ V is a divisor with normal crossings.”
- A new proof of desingularization over fields of characteristic zero. (2001, background): “We finally refer to [10] for an extension of this proof to the class of locally embedded excellent schemes; and for a full description of this equivariant algorithm in this more ample context.”