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Suppose that $C \rightarrow D$ is a finite morphism of projective schemes over an algebraically closed field of characteristic 0. The morphism is not birational, and $C$ and $D$ are reducible but reduced. If we take the Galois closure $C' \rightarrow C \rightarrow D$ then $C' \rightarrow C$ is a finite birational morphism of projective schemes. Is this morphism a blowup?

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  • $\begingroup$ If $C\to D$ is the normalisation of a cuspidal curve, what is $C'$? Don't you need some extra hypotheses for this question to make sense? $\endgroup$ – Kevin Buzzard Jan 20 '17 at 18:29
  • $\begingroup$ Please see my response below. $\endgroup$ – Jeremy Berquist Jan 20 '17 at 18:37
  • $\begingroup$ Experience shows that this site works best if you make your remarks and clarifications by editing the question (where everyone will see them) rather than as comments to answers. In particular, it's best to ask exactly what you want to ask in the question itself (and clarifying what extra assumptions you are and are not happy to make). $\endgroup$ – Kevin Buzzard Jan 20 '17 at 19:30
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Liu, Theorem 8.1.24: Let $f: Z \to X$ be a projective birational morphism of integral schemes. Suppose $X$ is quasi-projective over an affine Noetherian scheme. Then $f$ is the blowing-up morphism of $X$ along a closed subscheme.

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    $\begingroup$ In response to the answer given above, what if the varieties are reducible? $\endgroup$ – Jeremy Berquist Jan 20 '17 at 18:34
  • $\begingroup$ I would check Hartshorne too. There is some result along the lines of 'any projective morphism is the blow up of some coherent sheaf of ideals'. I don't remember the conditions on the schemes thought. $\endgroup$ – meh Jan 21 '17 at 0:56
  • $\begingroup$ This was meant as an answer to your second question: "If we take the Galois closure $C' \to C \to D$ then $C' \to C$ is a finite birational morphism of projective schemes. Is this morphism a blowup?" $\endgroup$ – user19475 Jan 21 '17 at 8:22

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