A Simplified Proof of Desingularization and Applications

This paper contains a short and simplified proof of desingularization overfields of characteristic zero, together with various applications to otherproblems in algebraic geometry (among others, the study of the behavior ofdesingularization of families of embedded schemes, and a formulation ofdesingularization which is stronger than Hironaka's). Our proof avoids the useof the Hilbert-Samuel function and Hironaka's notion of normal flatness: Firstwe define a procedure for principalization of ideals (i.e. a procedure to makean ideal invertible), and then we show that desingularization of a closedsubscheme $X$ is achieved by using the procedure of principalization for theideal ${\mathcal I}(X)$ associated to the embedded scheme (X). The paperintends to be an introduction to the subject, focused on the motivation ofideas used in this new approach, and particularly on applications, some ofwhich do not follow from Hironaka's proof.