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Let $k$ be a field and $X$ be a projective variety over $k$. I think it should be true that if $L$ is a line bundle on $X$ such that exists $s \in \Gamma(X,L)$ with $s_x \neq 0$ for all $x \in X$, then $L$ is isomorphic to $\mathcal{O}_X$. At least this is what analytic intuition would tell me. I am looking for a good proof or reference of this fact.

user110071
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There is a basic correspondence between sections of $L$ and maps $\mathcal O_X \to L$ (this is basically the $R$-module isomorphism $M\cong Hom_R(R,M)$ given by $m \mapsto [1 \mapsto m]$ sheafified). In this case, $\mathcal O_X \to L$ is given by $1\mapsto s$. Since $s$ is nonzero in every stalk, it globally generates $L$, so this sheaf map induces an isomorphism $\mathcal O_{X,x} \to L_x$ by $1_x\mapsto s_x$ on every stalk (this is an isomorphism on each fiber since it is a nonzero linear map of one-dimensional $k(x)$-vector spaces, hence an isomorphism on stalks by Nakayama's lemma). And of course a sheaf map which induces an isomorphism on stalks is itself an isomorphism.

Tabes Bridges
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