Desingularization algorithms I. Role of exceptional divisors
  The article is about a "desingularization principle" common to various canonical desingularization algorithms in characteristic zero, and the roles played by the exceptional divisors in the underlying local construction. We compare algorithms of the authors and of Villamayor and his collaborators, distinguishing between the fundamental effect of the way the exceptional divisors are used, and different theorems obtained because of flexibility allowed in the choice of "input data". We show how the meaning of "invariance" of the desingularization invariant, and the efficiency of the algorithm depend on the notion of "equivalence" of collections of local data used in the inductive construction. 
Edward Bierstone and Pierre D. Milman