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)inmathbbA3:xy−z2=0
in mathbbA3. Then you have the obvious map phicolonmathbbA2toC. But now suppose that C′ is a resolution of C provided by a putative "resolution functor". Then if we let tildeC be the minimal resolution, C′ factors through C. If we assume that mathbbA2 is resolved by itself (as seems reasonable!) then we'd have to have phi lifting to a map mathbbA2totildeC 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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