Desingularization algorithms I. Role of exceptional divisors

TL;DR

This paper analyzes various canonical desingularization algorithms in characteristic zero, focusing on the role of exceptional divisors and how their treatment affects invariance and efficiency.

Contribution

It compares different algorithms, clarifies the role of exceptional divisors, and explores how choices in local data influence the desingularization process.

Findings

Comparison of algorithms by authors and Villamayor et al.

Role of exceptional divisors in local construction clarified.

Invariance and efficiency depend on data equivalence notions.

Abstract

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.