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My question refers to a example from "Vector Bundles on Complex Projective Spaces" by Christian Okonek, Michael Schneider, Heinz Spindler (page 16):

Here $C_i \cong \mathbb{P}^1$ and $N_{C_i/X}$ is the notation for dual of the conormal bundle $N_{C_i/X} ^*:= J_{C_i}\vert_{C_i}$ where $J_{C_i}$ is corresponding to the ideal sheaf of $C_i$ via exact sequence

$$0 \to J_{C_i} \to \mathcal{O}_X \to \mathcal{O}_{C_i} \to 0$$

(compare with page 2)

Therefore that suffice to show that $N_{\mathbb{P}^1/X} ^* = \mathcal{O}_{\mathbb{P}^1}(1)$ holds, but why?

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This is due the following two facts (I'm not very consistent with my notation .. the notation for each line are independent):

1) Whenever you blow up a closed subscheme $Y\subset X$ the exceptional divisor $E$ is an effective cartier divisor on the blow up $\tilde{X}$ whose corresponding invertible sheaf is given by $\mathcal{O}_{\tilde{X}}(-1)$.

2) Given an effective cartier divisor $D$ in $X$, the normal bundle is given by $\mathcal{O}_X(D)|_D$.

Aside: Note also that $\mathcal{O}_{C_i}(-1)$ is not $\mathcal{O}_{\mathbb{P}^1}(-1)$, as you seem to be suggesting. Rather, $\mathcal{O}_{C_i}(-1) = \mathcal{O}_{C_i} \otimes i^*\mathcal{O}_{X}(-1)$ (where here I'm following the notation in the text you quoted).

You may find Vakil's chapter on blow ups helpful.

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