Abstract
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.
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References
D. Abramovich, M. Temkin, and J. Wlodarczyk. Functorial embedded resolution via weighted blowing ups. arXiv:1906.07106 (2019).
A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. Inst. Hautes Études Sci. Publ. Math. (1964-1967), no. 20 (§0–1), 24 (§2–7), 28 (§8–15), 32 (§16–21), p. 1101, Rédigés avec la collaboration de Jean Dieudonné.
A. Grothendieck. Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin, 1971, Séminaire de géométrie algébrique du Bois Marie 1960–61, Augmenté de deux exposés de M. Raynaud.
J. Kollár. Lectures on Resolution of Singularities, Annals of Mathematics Studies, Vol. 166. Princeton University Press, Princeton, NJ (2007).
H. Matsumura. Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8. Cambridge University Press, Cambridge (1986), Translated from the Japanese by M. Reid.
M. McQuillan. Semi-Stable Reduction of Foliations, Revised version of IHES pre-print, http://www.mat.uniroma2.it/~mcquilla/files/newss.pdf (2017).
M. McQuillan and D. Panazzolo. Almost étale resolution of foliations. Journal of Differential Geometry (2)95 (2013), 279–319
D. Panazzolo. Resolution of singularities of real-analytic vector fields in dimension three. Acta Mathematica (2)197 (2006), 167–289
G. Scheja and U.Storch. Differentielle eigenschaften der lokalisierungen analytischer algebren. Mathematische Annalen (2)197 (1972), 137–170
O. Villamayor. Equimultiplicity, algebraic elimination, and blowing up. Advances in Mathematics 262 (2014), 313–369
A. Vistoli. Intersection theory on algebraic stacks and on their moduli spaces. Inventiones Mathematicae (3)97 (1989), 613–670
B. Youssin. Newton polyhedra without coordinates. Mem. Amer. Math. Soc. (433)87 (1990), i–vi, 1–74
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McQuillan, M. Very functorial, very fast, and very easy resolution of singularities. Geom. Funct. Anal. 30, 858–909 (2020). https://doi.org/10.1007/s00039-020-00523-7
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DOI: https://doi.org/10.1007/s00039-020-00523-7