Is Hironaka's resolution of singularities functorial? I know that the resolution is not unique, there are flips etc. But if we have a rational map f:X---> Y, can we chose resolutions X'->X and Y'->Y and a map $f_*:X'\to Y'$ that makes the relevant diagram commute?
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In addition to what Damiano says, characteristic zero resolution is canonical in a stronger sense: a readable account is given in a paper by Hauser. In particular, resolution commutes with smooth morphisms and commutes with group actions. Recently, this has also been shown to be true for all excellent reduced schemes of dimension at most two by Cossart, Jannsen, and Saito |
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A useful (at least for me) example is given in Kollar's article/book on resolutions of singularities about how you can't expect to get a "resolution functor": take a quadric cone $$C = {(x,y,z) \in \mathbb A^3: xy-z^2=0}$$ in $\mathbb A^3$. Then you have the obvious map $\phi\colon \mathbb A^2 \to C$. But now suppose that $C'$ is a resolution of $C$ provided by a putative "resolution functor". Then if we let $\tilde{C}$ be the minimal resolution, $C'$ factors through $C$. If we assume that $\mathbb A^2$ is resolved by itself (as seems reasonable!) then we'd have to have $\phi$ lifting to a map $\mathbb A^2 \to \tilde{C}$ compatibly with the original morphism, which of course one cannot do. I found the introduction to Kollar's article really useful in understanding what one can and cannot expect from resolution of singularities. |
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The original theorem proved by Hironaka was not functorial. That feature was added later. |
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