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.