Let $P$ be a point in $\mathbb{P}^n$. Then we have a regular map $\mathbb{P}^n\setminus P\rightarrow \mathbb{P}^{n-1}$, which is not defined only at $P$. Then We know that the graph closure of this map is the Blow Up of $\mathbb{P}^n$ at $P$. $Blow\_P \mathbb{P}^n\hookrightarrow \mathbb{P}^n\times \mathbb{P}^{n-1}$. Now Let $p\_2$ be the projection to the second factor. $p\_2:Blow\_P \mathbb{P}^n\rightarrow \mathbb{P}^{n-1}$. My question is what is $p\_2^\*\mathcal{O}(1)$? Is it $\mathcal{O}(-E)$ or $p\_1^\*\mathcal{O}(1)\otimes \mathcal{O}(-E)$?


Here $E$ is the exceptional divisor and $p\_1$ is the projection onto the first factor.



Ample line bundle of Blow up
You can undertstand this by thinking about the simplest case $n=2$, the case $n \geq 3$ being analogous.


The blow-up of $\mathbb{P}^2$ at a point $P$ is the Hirzebruch surface $\mathbb{F}\_1$, and the two projections $$p\_1 \colon \mathbb{F}\_1 \longrightarrow \mathbb{P}^2, \quad p\_2 \colon \mathbb{F}\_1 \longrightarrow \mathbb{P}^1$$
are the blow-up morphism and the $\mathbb{P}^1$-fibration induced by the pencil of lines through $P$, respectively. So, denoting by $f$ the fibre of $p\_2$, we have $$p\_1^\*\, \mathcal{O}(1) = E+f, \quad p\_2^\* \,\mathcal{O}(1) = f,$$
that is $p\_2^\*\,\mathcal{O}(1)=p\_1^\*\,\mathcal{O}(1)\otimes \mathcal{O}(-E).$


