# A simplified proof of desingularization and applications.

Ana María Bravo; Santiago Encinas; Orlando Villamayor Uriburu

Revista Matemática Iberoamericana (2005)

- Volume: 21, Issue: 2, page 349-458
- ISSN: 0213-2230

## Access Full Article

top## Abstract

top## How to cite

topBravo, Ana María, Encinas, Santiago, and Villamayor Uriburu, Orlando. "A simplified proof of desingularization and applications.." Revista Matemática Iberoamericana 21.2 (2005): 349-458. <http://eudml.org/doc/41939>.

@article{Bravo2005,

abstract = {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. },

author = {Bravo, Ana María, Encinas, Santiago, Villamayor Uriburu, Orlando},

journal = {Revista Matemática Iberoamericana},

keywords = {Geometría algebraica; Singularidades; Geometría birracional},

language = {eng},

number = {2},

pages = {349-458},

title = {A simplified proof of desingularization and applications.},

url = {http://eudml.org/doc/41939},

volume = {21},

year = {2005},

}

TY - JOUR

AU - Bravo, Ana María

AU - Encinas, Santiago

AU - Villamayor Uriburu, Orlando

TI - A simplified proof of desingularization and applications.

JO - Revista Matemática Iberoamericana

PY - 2005

VL - 21

IS - 2

SP - 349

EP - 458

AB - 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.

LA - eng

KW - Geometría algebraica; Singularidades; Geometría birracional

UR - http://eudml.org/doc/41939

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.