Hartshorne Exercise II.7.8.
Let $X$ be a noetherian scheme, let $\mathcal E$ be a coherent locally free sheaf on $X$, and let $\pi: \Bbb P(\mathcal E)\ra X$ be the corresponding projective space bundle. Show that there is a natural 1-1 correspondence between sections of $\pi$ (i.e., morphisms $\sigma: X\ra \Bbb P(\mathcal E)$ such that $\pi\circ\sigma=\op{id}_X$) and quotient invertible sheaves $\mathcal E\ra\mathcal L\ra 0$ of $\mathcal E$.
Given a section $\sigma: X\ra \Bbb P(\mathcal E)$ of $\pi:\Bbb P(\mathcal E)\ra X$ we obtain a surjection of sheaves $\mathcal E\ra \mathcal L\ra 0$ as follows. Recall that there is a natural surjection $\pi^*\mathcal E\ra \OO_{\Bbb P(\mathcal E)}(1)$. Also consider Proposition II.7.12 of Hartshorn, which says for any morphism $g:Y\ra X$, the morphisms $Y\ra \Bbb P(\mathcal E)$ over $X$ are in 1-1 correspondence with line bundles $\mathcal L\in \op{Pic}Y$ and a surjection $g^*\mathcal E\ra \mathcal L\ra 0$ of sheaves on $Y$. In considering the scheme $X=Y$ and the morphism $\pi\circ\sigma=\op{id}_X:X\ra X$, it follows immediatly that for each section $\sigma$ there is a unique line bundle $\mathcal L$ on $X$ that is a quotient of $\mathcal E$, because morphisms $X\ra \Bbb P(\mathcal E)$ over $X$ are precisely sections $\sigma$ of $\pi$. Moreover, proposition II.7.12 gives the explicit construction for $\mathcal L$ as $\mathcal L=\sigma^*\OO_{\Bbb P(\mathcal E)}(1)$.
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