Desingularization of embedded excellent surfaces

Resolution of singularities of threefolds in positive characteristic. I.

“…In the case of proposition when I is locally principal, see [8] (a self contained proof in the most general case where Z := V (I) is a reduced closed subscheme of any excellent regular scheme of dimension three is in preparation, work of the first author in collaboration with U. Janssen and S. Saito). Reducing to that case is the content of Proposition 4.2; we include a guide of proof of Proposition 4.2 for self-completeness of this article.…”

Resolution of singularities for 3-folds in positive characteristic

“…Some details on this are given in [H4], [H5], [H6] and [Co1]. A complete proof of resolution of excellent surfaces using a different method, is given by Lipman in [L2].…”

“…Hironaka announced in his notes [H2] a proof of resolution of singularities of an arbitrary excellent surface. In the notes he gives a few comments about the realization of the general proof, and gives some parts of the proof in [H3], [H4] and [H5] (see also Cossart's paper [Co1]). We outline this approach, when S is a surface which is embedded in a nonsingular variety V of arbitrary dimension d, over an algebraically closed field k.…”

“…Some papers making progress on resolution in positive characteristic are Cossart [Co1], [Co2], [Co3], [Co4], Giraud [G], Hauser [Hau1], de Jong [Jo], Hironaka [H9], Kawanoue [Ka], Kawanoue and Matsuki [KM], Kuhlmann [Ku], Moh [M], Piltant [P], Spivakovsky [S] and Teissier [T]. Several simplifications of Hironaka's proof of characteristic zero resolution have appeared, making the proof quite accessible now, including Abramovich and de Jong [AJo], Bierstone and Milman [BrM], Bogomolov and Pantev [BP], Bravo, Encinas and Villamayor [BEV], Encinas and Hauser [EH], Encinas and Villamayor [EV], Hauser [Hau2], Kollár [Ko], Villamayor [V], Wlodarczyk [W].…”

Two Computational Techniques for Singularity Resolution

“…The interested reader might also wish to consult the following papers on the topic: Hironaka (1984) presenting a constructive characteristicfree proof of resolution of surface singularities, followed by a generalization for nonalgebraically closed base fields in Cossart (1981) and a conceptualization and globalization in the context of taut resolution in Hauser (1998); providing a survey on the difficulties of resolution of singularities, especially in positive characteristic; Encinas and Hauser (2000) presenting a simplified proof for constructive resolution of singularities after Villamayor and Bierstone-Milman;de Jong (1996) with a characteristic-free approach via alterations.…”