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Let $X = \mathbb{P}^{n}$ and $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ along a subvariety $Y$ with exceptional divisor $E$.

According to the following answer in mathoverflow Blowing-up and direct image sheaf, I have the following questions

1) In this case, it is true that $\pi_{*}(\mathcal{O}_{\widetilde{X}}(-mE)) = I_{Y/X}^{m}$ for $m\geq 1$?

2) I can't understand the answer from the point, Taking Proj this means that there is an embedding... (see last answer please)

I am using this result for $X = \mathbb{P}^{n} $ but, honestly, I don't understand the answer, for sure, because I am new to the subject. Could someone help me with this?

Thank you very much.

red_trumpet
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Emanuell
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    If your $Y$ is smooth, the above should answer your question. If $Y$ is not smooth, then things can go very wrong. (Your assumption that $X=\Bbb P^n$ says that $X$ is smooth, which removes the other place things can go wrong.) – KReiser Jan 15 '20 at 03:31
  • @KReiser. Thanks for your suggestion. I will read the answer. – Emanuell Jan 15 '20 at 10:23
  • @KReiser. If I have any questions, can I ask here in the comments for you? Thank you. – Emanuell Jan 15 '20 at 10:27
  • @red_trumpet. Thanks for the edit. – Emanuell Jan 15 '20 at 10:30
  • If you have specific questions about my answer, you should put them in this or another question - comments aren't really for big discussions on this site. – KReiser Jan 15 '20 at 10:39
  • @KReiser. Ok. I will turn the doubts into a new question. Thanks for the tip. – Emanuell Jan 15 '20 at 11:00

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