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?
> 
> 
> 



Hironaka desingularisation theorem -- new proofs in literature?
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)](https://doi.org/10.1090/S0273-0979-03-00982-0)* (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.


