Basic Notions: Michael Tenkin "Resolution: classical, relative and weighted"



Until recently there was known an essentially unique way to resolve singularities of varieties of 
characteristic zero in a canonical way (though there were different descriptions and proofs of correctness). 

Recently a few advances happened -- 
1) The algorithm was extended to varieties with log structures and even to morphisms -- This requires to consider 
more general logarithmic blow ups and results in a stronger logarithmic canonicity of the algorithms even for resolution 
of usual varieties. However, one also has to consider certain weighted blow ups and work with orbifolds. 
(joint work with Abramovich and Wlodarczyk)

2) Probably a simplest algorithm for resolving varieties was found -- it does not use history or normal crossing divisors at all, 
and simply improves a basic invariant at each blow up. This requires to work with general weighted  blow ups, and one has 
to work with orbifolds. (joint work with Abramovich and Wlodarczyk, was discovered independently by McQuillan)

3) A combination of 1) and 2) should be possible too. (For resolution of log varieties this was done by Quek Ming Hao.)
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