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?



Functoriality of Hironaka's resolution of singularities
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.](http://www.ams.org/bull/2003-40-03/S0273-0979-03-00982-0/S0273-0979-03-00982-0.pdf) 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](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2191v1.pdf)


