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Resolution in positive characteristic?

Posted by Charles Siegel under

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[4] Comments
Ok, rumor has it that Hironaka is claiming a proof of resolution of singularities in positive characteristic. Anyone know anything more about this? Do we have any readers at Harvard that can confirm this rumor or squash it? Do I need to haul myself to Boston on the double? Please, news!

Source: Not Even Wrong

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September 24, 2008 at 7:39 am

I didn’t go to the talk, but from what I’ve heard it didn’t sound like a big momentous talk. They scheduled it at 9 AM, presumably to discourage people from coming just to see Hironaka.

September 24, 2008 at 9:08 am

Thanks Eric. A commenter on Not Even Wrong has also commented that Hironaka isn’t claiming a proof, just a new approach. Well, that’s good too, though. Sorry if I got peoples’ hopes up on an unfounded rumor.

September 25, 2008 at 1:53 pm

Hi,

if X is a reduced, noetherian, excellent scheme of dimension two, then it exists a strong desingularization. This was proven by Lipman (Desingularization of two-dimensional schemes, Ann. Math. 107

(1978), 151-207). For the definition of “excellent” see Matsumura “Commutative Ring theory”.

With some extra work you get another nice result:

Let X->S be a fibered surface over a Dedekind scheme with dim S = 1 and smooth generic fiber. Then X admits a strong desingularization.

You can find this result in Liu’s excellent book “Algebraic geometry and arithmetic curves” wich is a mix of Hartshorne and EGA.

Best regards!

September 25, 2008 at 6:35 pm

I attended a talk by Hironaka on positive characteristic resolution about 6 or 7 years ago. He was explicit at the beginning of the talk that he had not solved resolution at that time. So the fact that there is confusion among some participants about whether he claimed resolution in his recent talk is interesting in itself. There will be a RIMS meeting this December on resolution, and probably any confusion will be settled after that.