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.
"Some natural properties of constructive resolution of singularities." Asian J. Math. 15 (2) 141 - 192, June 2011.