Poll about your proof of resolution of singularities and a request for advice

Question

The questions first: What is the proof of resolution of singularities that you know?

Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of characteristic zero, all with more or less similar flavor but different in technical details and in choices that the resolution algorithm allows us to make. When writing a proof that uses specific features of some of those details I can't stop being uneasy about assuming the reader read about the specific constructions elsewhere. I would like to know from MOers what proof you have seen and if you have a reason for the choice, if it was a choice, I would like to hear it too.

Maybe asking about what you know is too invasive. I am just asking for the proof that you happened to find in your way, even you have only read a few lines of it.

The purpose of the question: The conspicuous one. To get a sense, by a rough approximation and a small sample, of what proofs are more culturally known. Have a concrete feeling when sending a reader to find the details in other paper, either of feeling OK with it or of guilt.

It is a question about fashion, which also has its role in mathematics... and knowing what the fashion is is useful.

What details?: Although I had in mind a specific detail of the proofs I didn't mention it because it is not the only one that changes from proof to proof and because the result of the poll gives information about all of them. Examples are: the resolution invariant, the ways of making the local descriptions of the centers of blowings-up match to form a globally defined smooth subvariety, the ways of getting functoriality and the different meanings that functoriality can have...

(edit) Forgot the "request for advise": If you have would like to give advise about how you have dealt with similar situations and describe your example that is welcomed.

It is a community wiki question, so feel free to change what is said here if needed or if you want the poll to also give information about other questions that you would like to be answered. (or for correcting the English!)

Answer 1

All right. Since you ask, the only treatment of resolution of singularities that I have even glanced at is Herwig Hauser's 2003 Bulletin article:

http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/S0273-0979-03-00982-0.pdf

I can't remember how much of it I ever read -- certainly less than half -- but I do remember enjoying the article when it came out, getting something out of it, and finding the pictures attractive and better-than-usually integrated into the text. (I am not a very visual thinker; when someone shows a fancy color portrait in their talk and claims it is a cross section of a K3 surface -- or whatever -- I wonder whether I am being put on.)

Because the article is freely available online, it would seem to make a good reference, provided of course that it contains the information you want to cite.

Answer 2

Several years ago I looked carefully at several of these. The first that I looked carefully is:

Simple Hironaka resolution in characteristic zero by Wlodarczyk. This article has the advantage that it is self-contained and complete and relatively short. It lacks discussion of examples in higher dimensions and motivation that other papers have, as it is quite to the point.

I also read A Simplified Proof of Desingularization and Applications by Bravo, Encinas and Villamayor which has a lot more discussion (it's about 100 pages instead of about 30).

There is also Kollár's quite recent book which I have only skimmed, and I've read large parts of Bierstone and Milman's paper. There have been a number of simplifications and improvements over the past decade (and a little more) so the more modern accounts are certainly more streamlined (and give better results in some directions).

I should point out that many of the more modern proofs incorporate ideas from each other, making the proof more and more streamlined.

Answer 3

As a bit of an update on the subject in characteristic 0, there have been some more recent papers which try to vary the desingularization algorithm slightly to achieve a better resolution of singularities, for example algorithms which avoid simple normal crossings or semi-simple normal crossings or even "minimal" singularities. These proofs shed light on the original algorithm while showing how far we can push this approach. If you read through some of these articles, you will not only get a glimpse of the original algorithm, but you will see how the desingularization invariant can be used as a computational tool. Here are the more recent articles on the topic:

• Bierstone/Da Silva/Milman/Pacheco, Desingularization by blowings-up avoiding simple normal crossings

• Bierstone/Pacheco, Resolution of singularities of pairs preserving semi-simple normal crossings

• Bierstone/Pacheco, Desingularization preserving stable simple normal crossings

• Bierstone/Milman, Resolution except for minimal singularities 1 and Bierstone/Lairez/Milman, [...] 2 (the case of four variables)

I found the following two articles very enlightening when I was learning about the topic:

• Bierstone/Milman, Resolution of singularities

• Bierstone/Milman, Functoriality in resolution of singularities

The first goes into great detail about some of the subtle points of the algorithm, while the second gives a rather concise proof of resolution of singularities.

Finally, the article below is quite distinct in that it is the first time (as far as I know) that the complexity of the desingularization algorithm has been studied - that is, how many steps in the algorithm occur before it terminates? Can we find an upper bound? You can read more about that here:

• Bierstone/Grigoriev/Milman/Wlodarczyk, Effective Hironaka resolution and its complexity (with appendix on applications in positive characteristic)