Resolution in characteristic 0 using weighted blowing up

This is joint work with Michael Tëmkin (Jerusalem) and Jarosław Włodarczyk (Purdue), a side product of our work on functorial semistable reduction. A similar result was discovered by G. Marzo and M. McQuillan. Given a variety $X$, one wants to blow up the worst singular locus, show that it gets better, and iterate until the singularities are resolved. Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci – it goes into a loop. We show that there is a functorial way to resolve varieties using weighted blowings up, in the stack-theoretic sense. To an embedded variety $X \subset Y$ one functorially assigns an invariant $(a_1,\ldots,a_k)$, and a center locally of the form $(x_1^{a_1} , \ldots , x_k^{a_k})$, whose stack-theoretic weighted blowing up has strictly smaller invariant under the lexicographic order.

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