# Zariski's Main Theorem and Blow up

Let $$X$$ and $$Y$$ be smooth projective schemes with $$Y \subset X$$. Let $$\pi : \widetilde{X} \to X$$ be the blow up of $$X$$ along $$Y$$ with exceptional divisor $$E$$.

I have seen the statement that Zariski's Main Theorem implies that $$\pi_{*}(\mathcal{O}_{\widetilde{X}} ) \to \mathcal{O}_{X}$$ and $$\pi_{*}(\mathcal{O}_{E}) \to \mathcal{O}_{Y}$$ are isomorphisms. Why is this true?

References and suggestions will be appreciated.

This MO post is a great reference if you're ever trying to figure out or remember when for a morphism $$f:X\to Y$$ we will have $$f_*\mathcal{O}_X=\mathcal{O}_Y$$.

Here's the relevant portion of that answer for this post, in order to make this answer self-contained:

The case of an arbitrary projective morphism.

Now when $$f:X\to Y$$ is any projective morphism, then $$f_*\mathscr O_X$$ is a coherent $$\mathscr O_Y$$-module, hence we get a factorization of $$f$$ as $$h\circ g:X\to Z\to Y$$, where $$h:Z\to Y$$ is affine, and where also $$h_*(\mathscr O_Z) = f_*\mathscr O_X$$. Then $$h$$ is not only an affine map, but since $$h_*(\mathscr O_Z)$$ is a coherent $$\mathscr O_Y$$-module, $$h$$ is also a finite map. Moreover $$g:X\to Z$$ is also projective and since $$g_*(\mathscr O_X) = \mathscr O_Z$$, it can be shown that the fibers of $$g$$ are connected. Hence an arbitrary projective map $$f$$ factors through a projective map $$g$$ with connected fibers, followed by a finite map $$h$$. Thus in this case, the algebra $$f_*\mathscr O_X$$ determines exactly the finite part $$h:Z\to Y$$ of $$f$$, whose points over $$y$$ are precisely the connected components of the fiber $$f^{-1}(y)$$.

One corollary of this is "Zariski's connectedness theorem". If $$f:X\to Y$$ is projective and birational, and $$Y$$ is normal then $$f_*\mathscr O_X= \mathscr O_Y$$, and all fibers of $$f$$ are connected, since in this case $$Z = Y$$ in the Stein factorization described above. If we assume in addition that $$f$$ is quasi finite, i.e. has finite fibers, then $$f$$ is an isomorphism. More generally, if $$Y$$ is normal and $$f:X\to Y$$ is any birational, quasi - finite, morphism, then $$f$$ is an embedding onto an open subset of $$Y$$ ("Zariski's 'main theorem' "). More generally still, any quasi finite morphism factors through an open embedding and a finite morphism.

This applies to your situation as follows: the blowup map $$\pi:\widetilde{X}\to X$$ is a projective birational map with connected fibers. Since projective and connected fibers are preserved under base change, we see that the base change of this map $$E\to Y$$ is again projective with connected fibers, so we may also apply the result there, via the Stein factorization described in the first paragraph (even though this last morphism is not birational - $$\dim E=\dim X-1\neq \dim Y$$).

• Hi, KReiser @. Thank you very much for the support given in the questions I ask here in the forum. Jan 17, 2020 at 11:34
• KReiser@. I asked this question because I am trying to understand your answer and another about Direct Image and Blow up. Jan 17, 2020 at 11:38
• Hi, @KReiser. I will post a question to answer two questions I have about your answer in Direct Image and Blow up. thank you. Jan 17, 2020 at 13:13