# Is the modification a rational map?

Good morning,

I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions.

We have the following theorem of Hironaka: Let $X\subset \mathbb{CP}^n$ a closed complex projective variety. Then, there exists a modification $f\colon (\tilde{X},E) \to (X, Sing(X))$ such that $\tilde{X}$ is smooth and projective, and the exceptional divisor $E$ is a divisor with normal crossings.

My questions: Are $f$ and $f^{-1}$ rational maps?

Any help is appreciated. Thanks in advance,

Duc Anh

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Yes, they are. $f$ is given by a sequence of blow ups, so in fact, it is a regular map (or morphism if you prefer). The inverse is, however, usually only rational. –  Donu Arapura Mar 17 '13 at 1:58
Thank you very much. –  Đức Anh Mar 17 '13 at 2:12