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 , 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.
Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ a long of $Y \subset X$, with exceptional divisor $E$ and $\text{dim}Y &gt; 0$,   where $X$ and $Y$ are smooth projectives schemes.   For an answer to a question asked here in the forum (), Blowing up and direct image of line bundle), we have: $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) = I_{Y/X} \tag{$**$}$$ here $I_{_{Y/X}}$ denotes the ideal sheaf of $Y$.  $(**)$ was obtained through the following exact sequence: $$0 \longrightarrow \mathcal{O}_{\widetilde{X}}(-E) \longrightarrow \mathcal{O}_{\widetilde{X}} \longrightarrow \mathcal{O}_{E} \longrightarrow  0$$  My doubt is: How to find $\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE)$ for $n \geq 0$? What if $n &lt; 0 $?  Thank you.
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