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I asked this at math.stackexchange, but received no reply.

Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.

Let $Y \to X$ be the normalization. The answer is positive in the following situations:

  • $X$ is quasi-projective. This follows since normalization is a projective birational morphism.
  • The singularities are mild, namely $X$ is Gorenstein and $Y$ is both Gorenstein and Cohen–Macaulay. In this case, by arXiv:1608.04525, normalization is the blowup along the conductor ideal.

By variety, I mean an integral separated scheme of finite type over the complex numbers.

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    $\begingroup$ It's true for affine $X$ by what you said, so it's true locally for any $X$. Blow ups patch, so it's true for $X$. $\endgroup$
    – Donu Arapura
    Commented Jun 1 at 14:21
  • $\begingroup$ @DonuArapura Using the local arguments that you suggest, the following result is proved in mathoverflow.net/a/53811: if $f\colon Z \to X$ is any projective birational morphism of varieties, then there exists a blowup $\pi$ such that $\pi$ factors through $f$. But I do not see how to do the gluing to directly conclude that $f$ is a blowup. $\endgroup$
    – SeparatedScheme
    Commented Jun 1 at 17:13

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