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Jan 21, 2017 at 8:22 comment added user19475 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?"
Jan 21, 2017 at 0:56 comment added meh 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.
Jan 20, 2017 at 18:34 comment added Jeremy Berquist In response to the answer given above, what if the varieties are reducible?
Jan 20, 2017 at 18:28 history answered user19475 CC BY-SA 3.0