I'm wondering what the landscape looks like for proofs of Hironaka's desingularisation theorem.
Are there many proofs in the literature?
Is there a commonly accepted simplest bare-knuckle proof out there that might be considered a pleasant read for people outside of algebraic geometry?
Would that still be Hironaka's original manuscript?
There's an article by Herwig Hauser in the Bulletin of the AMS: The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand) (Bull. Amer. Math. Soc. 40 (2003), 323-403 )
which is aimed at giving an accessible account (I must admit I didn't read most of it though).
Mircea Mustata also gave a 5 lecture course at the 2008 Park City summer school. He did not prove every fact he needed, but the lectures contained all of the relevant ideas. If I remember correctly, the main idea was to find the right invariants for varieties (I think it has something to do with multiplier ideals, but I could be wrong) and just show that one can make the invariants "smaller" by choosing the right blow-ups to do at each step. Someone correct me if I am wrong, please. I don't know if those notes can be downloaded anywhere, but they will appear in the summer school proceedings.
As far as I have been informed, the original proof of Hironaka is extremely difficult to understand.
you might be looking for Kollar's book Lectures on Resolution of Singularities
For a long time, Hironaka's paper was probably the canonical reference. New proofs started to crop up in the late 90's, and truly changed the perception of the problem. The motivation was to make Hironaka's proof more constructive, and ultimately, make it tractable by computer algebra. This was ultimately done by Schicho and Bodnar in Maple (desing project). (Of course, the complexity is atrocious, but it still allows to run example beyond what one could reasonably do by hand.)
Two main groups worked on it in the late 90's, and they have been referenced already in the prior answers: Bierstone-Milman on the one hand, and Encinas-Villamayor and maybe a couple of others on the other hand (spanning many papers). There was a year-long seminar at Purdue in 2000-2001 led by Kenji Matsuki and Andrei Gabrielov to work through the Encinas-Villamayor proof. This was an opportunity to clarify quite a few things in the construction, and Matsuki posted his notes (128 p.) on Arxiv.
Ultimately, this lead Jaroslaw Wlodarczyk to offer the shortest proof of desingularization (by far, 24 pages vs. 100+ for everyone else) .
Jaroslaw Wlodarczyk
Simple Hironaka resolution in characteristic zero.
J. Amer. Math. Soc. 18 (2005), no. 4, 779--822
Arxiv
I hope this helps! (Sorry for the delay in my answer, I only joined MO last week-end.)
(Added later) How could I forget? There is also a nice expository paper that covers all these developments. It should be very useful for someone trying to understand how the various works fit together (does predate Wlodarczyk's paper though).
H. Hauser
The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand).
Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403
There are some very short proofs of resolution using toric geometry and de Jong's idea of semistable reduction of curves that are independent of Hironaka's work. These proofs seem to be largely ignored by the resolution community, possibly because it is not clear that the resolutions they produce have various nice properties such as functorialiy for smooth morphisms.
See
Smoothness, Semistability, and Toroidal Geometry by Dan Abramovich, Johan de Jong
Weak Hironaka theorem by Fedor Bogomolov, Tony Pantev
for a couple of 10 page proofs using these ideas (though both proofs assume deep results about semistable reduction and toric geometry, so maybe the short length is a bit misleading).
When I was learning this, I found several articles by Villamayor et al to be very illuminating.
In particular, see the following paper of Bravo, Encinas and Villamayor: A Simplified Proof of Desingularization and Applications, Rev. Mat. Iberoamericana 21 (2005), no. 2, 349-458.
There is also Dale Cutkosky's book, Resolution of Singularities.
More in line with the main question, I'll point out that Bierstone and Milman have "A simple constructive proof of Canonical Resolution of Singularities," which may or may not be simple, but is certainly constructive, unlike Hironaka's.
Bierstone E., Milman P.D. (1991) A simple constructive proof of Canonical Resolution of Singularities, In: Mora T., Traverso C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. doi:10.1007/978-1-4612-0441-1_2.
:)
Old thread, but let me point out the recent preprint Very fast, very functorial, and very easy resolution of singularities, by M. McQuillan and G. Marzo.
There is a also published version on GAFA, with McQuillan listed as the only author.
This may not be the answer to the question, but you may be interested. You'll find his recent talks.
http://www.ms.u-tokyo.ac.jp/video/conference/2009gcoeopsym/index.html
