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Let $X$ be a projective manifold, and $Z$ be a submanifold of $X$ with codimension at least 2. Let $Y$ be the blow-up of $X$ along $Z$ with the exceptional divisor $E$. Then $\pi_*:T_Y\rightarrow \pi^*T_X$ induces a rational map $\widetilde{\pi}:\mathbf{P}(T_Y)\dashrightarrow\mathbf{P}(T_X)$. The pull back of the line bundle $\mathcal{O}_{\mathbf{P}(T_X)}(1)$ by $\widetilde{\pi}$ is denoted by $\widetilde{\pi}^*\mathcal{O}_{\mathbf{P}(T_X)}(1)\in \text{Pic}(\mathbf{P}(T_Y))$.

Question: what is the difference between $\widetilde{\pi}^*\mathcal{O}_{\mathbf{P}(T_X)}(1)$ and $\mathcal{O}_{\mathbf{P}(T_Y)}(1)$?

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