These expository notes, addressed to non-experts, are intended to present some of Hironaka’s ideas on his theorem of resolution of singularities. We focus particularly on those aspects which have played a central role in the constructive proof of this theorem.

In fact, algorithmic proofs of the theorem of resolution grow, to a large extend, from the so called Hironaka’s fundamental invariant. Here we underline the influence of this invariant in the proofs of the natural properties of constructive resolution, such as: equivariance, compatibility with open restrictions, with pull-backs by smooth morphisms, with changes of the base field, independence of the embedding, etc.

Building upon works of Hironaka, Bierstone-Milman, Villamayor and Włodarczyk. We give an a priori estimate for the complexity of the simplified Hironaka algorithm.

We prove that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a $Q$-variety. The desingularization algorithm is therefore $Q$-universal or absolute in the sense that it is induced from its restriction to varieties over $Q$. As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties.