Sur le polyèdre caractéristique d'une singularité

Resolution of singularities of threefolds in positive characteristic. I.

“…We have L = K i (W/V ) except if s = f = p d = 1, e = l in (9). Therefore the lemma follows from Proposition 9.1 except if s = f = p d = 1, e = l in (9), which we assume from now on.…”

“…On the other hand, dealing with nonperfect ground fields induces technical difficulties which do not seem to have been systematically considered in any written program to this date. Some partial results in dimension three are already known [9,34], but no such general statement as that of Conjecture 3.1.…”

“…78]. Finally, note that when L/K is finite Galois, W j /V is immediate if and only if e = f = 1 in (9). See [29] for background and further scrutiny on immediate extensions whose factor p d in (9) is not trivial.…”

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

“…This last result suggests that embedded resolution of singularities should hold, at least for 3-dimensional schemes over perfect fields. More results concerning resolution of singularities in positive characteristic can be found in [1,3,7,16,25,26,[28][29][30]38,39,42,43,48].…”

Resolution of singularities for 3-folds in positive characteristic

“…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].…”