Some natural properties of constructive resolution of singularities (Q408450)

The article under review is an introduction to the theory of algorithmic resolution of singularities of algebraic varieties and local principalization of sheaves of ideals. The authors explain the basic notions and discuss some natural properties of these algorithms, such as compatibility with pull-backs via smooth morphisms, equivariance, changes of base field, etc. They emphasize the importance of some fundamental ideas of Hironaka on the subject, such as his ``fundamental invariant'' (a certain fraction, involving the order of an ideal) and a notion of equivalence (requiring equalities of certain closed sets, which are singular loci). The authors present a fairly complete proof a an algorithmic resolution theorem (in characteristic zero), which is a streamlined version (and simplification) of one developed by Villamayor several years ago explained, for instance, in Chapter 6 of [\textit{S. D. Cutkosky}, Resolution of singularities. Graduate Studies in Mathematics 63. Providence, RI: American Mathematical Society (AMS). (2004; Zbl 1076.14005)].