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Let $f:Y\rightarrow X$ be a proper birational morphism. Suppose that $X$ is normal and $Y$ is smooth. Let us write the largest open subset U of X such that $f^{-1}$ can be defined.

I want to show that $\operatorname{codim}(X\backslash U)\geq2$.

I think that I should use Zariski's main theorem but I don't know how to use. Thank you.

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  • $\begingroup$ You write that “$Y$ is normal.” However the title of your post suggests that $Y$ is smooth and $X$ is normal. $\endgroup$
    – Jason Starr
    Commented Sep 6, 2023 at 15:53
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    $\begingroup$ Oh I made a mistake. I fixed it. $\endgroup$
    – Irish
    Commented Sep 6, 2023 at 16:04
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    $\begingroup$ You do not need Zariski's Main Theorem for this. It follows already from the valuative criterion of properness. $\endgroup$
    – Jason Starr
    Commented Sep 6, 2023 at 18:34

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