# rational map becomes a morphism after blow-ups

In dimension 2, a rational map becomes a morphism after a sequence of blow-ups. Does this still hold in higher dimensions?

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In light of the comments under abx's answer, it might be a good idea to clarify the question a bit; in particular, do you require the blow-ups to be along smooth centers? – user5117 Nov 23 '13 at 22:21

Yes, but this is a very difficult result, due to Hironaka. To be precise : given a rational map $f:X --> Y$, there exists a birational morphism $b:\hat{X}\rightarrow X$, obtained as the composition of successive blown-up with smooth centers, such that $f\circ b$ is a morphism.
Actually, you do not need Hironaka at all. Let $X$ and $Y$ be projective schemes. Let $f$ be a morphism from a dense open subset $U$ of $X$ to $Y$. Let $\Gamma_f$ be the graph of $f$ inside $X\times Y$. Let $\widehat{X}$ be the Zariski closure of $\Gamma_f$. Then the projection, $b:\widehat{X}\to X$, is a projective, birational morphism. Thus, by a theorem in Chapter II, Section 7 of Hartshorne's "Algebraic Geometry", the morphism $b$ is (isomorphic to) the blowing up of $X$ along a coherent sheaf of ideals. – Jason Starr Nov 23 '13 at 13:03