21

Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:

Kawanoue, Hiraku, Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration, Publ. Res. Inst. Math. Sci. 43, No. 3, 819-909 (2007). ZBL1170.14012.

Urabe, Tohsuke, New Ideas for Resolution of Singularities in Arbitrary Characteristic.

Edit: Another recent talk by Hironaka (in Vienna).

Scott Morrison
  • 7,540
Thomas Riepe
  • 10,731
  • 3
    Since I know some of the people involved, I'd rather not comment, except to point out that some talks from about a year and a half ago are publicly available here http://www.kurims.kyoto-u.ac.jp/~kenkyubu/proj08-mori/. – Donu Arapura May 03 '11 at 19:11
  • 1
    Deqr Thomas, My view on the question in your comment below: Yes, it would be a bit of a disaster if resolution (and more generally semi-stable reduction) was false in char. p (or in mixed characteristic)! Regards, Matthew – Emerton May 19 '11 at 12:35
  • http://plone.mat.univie.ac.at/events/2011/tba-17 404 – Piotr Achinger May 12 '17 at 04:54
  • 3
    The link for Vienna's talk is broken. – Leo Alonso May 13 '17 at 10:29

2 Answers2

32

Also, I'd like to point out this paper of Hironaka...

http://www.math.harvard.edu/~hironaka/pRes.pdf

I haven't read the paper, and also I haven't heard anybody talking about it in the last weeks, which I find a little strange given the problem in question... has anybody here gone through the proof?

pozio
  • 599
  • 1
    @Turbo: the first page says 23 March 2017. But there might have been earlier versions. – R. van Dobben de Bruyn May 12 '17 at 06:12
  • 3
    If anyone happen to have read it, please take a look if you can answer my confusion here: https://mathoverflow.net/questions/269651/hironakas-proof-of-resolution-of-singularities-in-positive-characteristics – Henry.L May 13 '17 at 21:25
4

An (or some) additional very recent references on resolution of singularities in positive characteristic:

There is a recent (expository) article by H. Hauser

On the Problem of Resolution of Singularities in Positive Characteristic (Or: A proof we are still waiting for), Bull. Amer. Math. Soc. 2010, Vol. 47,1; p.1-30.

Available on his webpage, where one can also find some preprints around this subject. For example,

Wild Singularities and Kangaroo Points for the Resolution in Positive Characteristic

  • Thanks a lot! (wondering if it would be a catastrophe if such a resolution would not exist?) – Thomas Riepe May 03 '11 at 13:23
  • @Thomas Riepe: You are welcome. Unfortunately I cannot answer your additional question. My knowledge on the subject is not at all well-developed; I just happened to have heard a talk of Hauser in a general context, and subsequently browsed some of his expository writings. –  May 03 '11 at 13:27