Some natural properties of constructive resolution of singularities

TL;DR

This paper explains Hironaka's ideas on the resolution of singularities, emphasizing the constructive and algorithmic aspects, especially the role of Hironaka's fundamental invariant in ensuring natural properties.

Contribution

It highlights the natural properties of constructive resolution of singularities and the influence of Hironaka's fundamental invariant on these properties.

Findings

Constructive resolution is equivariant and compatible with open restrictions.

Resolution is compatible with pull-backs by smooth morphisms.

Resolution process is independent of the embedding and base field changes.

Abstract

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.