Let $X$ be a complex manifold and $L$ a line bundle on it. Define $Y:=\mathbb{P}(L\oplus \mathcal{O}\_X)$ be the projective bundle over $X$. Here is a statement I don't understand:



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> The summands $L$ and $\mathcal{O}\_X$ respectively determine divisors on $Y$, each of which is isomorphic to $X$.
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How can one define such divisors on $Y$? 



A question on a projective bundle $\mathbb{P}(L\oplus \mathcal{O}_X)$
This is a rather standard fact. 


Let $\mathscr{E}$ be a rank $2$ locally free sheaf on $X$ and let $\pi \colon \mathbb{P}(\mathscr{E}) \to X$ be the corresponding $\mathbb{P}^1$-bundle. Then there is a natural correspondence between sections $\sigma \colon X \to \mathbb{P}(\mathscr{E})$ of $\pi$ and suriections $\mathscr{E} \to \mathscr{L} \to 0$, where $\mathscr{L}$ is a line bundle on $X$. Such a correspondence is given by $\mathscr{L}= \sigma ^\* \mathscr{O}\_{\mathbb{P}(\mathscr{E})}(1)$.


So in your setting the two quotients $L$ and $\mathscr{O}\_X$ of $\mathscr{E}=L \oplus \mathscr{O}\_X$ determine two sections of $\pi \colon \mathbb{P}(\mathscr{E}) \to X$. Of course, any section is a divisor in $\mathbb{P}(\mathscr{E})$ which is isomorphic to $X$ via the projection $\pi$.


