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Very functorial, very fast, and very easy resolution of singularities
Geometric and Functional Analysis  (IF2.148),  Pub Date : 2020-06-13, DOI: 10.1007/s00039-020-00523-7
Michael McQuillan

The main proposition, Theorem 1.2, is the existence for excellent Deligne–Mumford champ of characteristic zero of a resolution functor independent of the resolution process itself. Received wisdom was that this was impossible, but the counterexamples overlooked the possibility of using weighted blow ups. The fundamental local calculations take place in complete local rings, and are elementary in nature, while being self contained and wholly independent of Hironaka’s methods and all derivatives thereof, i.e. existing technology. Nevertheless Abramovich et al. (Functorial embedded resolution via weighted blowing ups, 2019. arXiv:1906.07106), have varied existing technology to obtain even shorter proofs of all the main theorems in the pure dimensional geometric case. Excellent patching is more technical than varieties over a field, and so easier geometric arguments are pointed out when they exist.