Wednesday, 23 May 2012

ag.algebraic geometry - Functoriality of Hironaka's resolution of singularities

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:xyz2=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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