# Non zero global section of line bundle and its dual $\implies \mathcal L\cong O_X$

Let $$X$$ be a projective integral scheme over $$K=\bar{K}$$. Suppose that $$\Gamma(X,\mathcal L)\neq 0$$, $$\Gamma(X,\mathcal L^{*})\neq 0$$. We want to show that $$\mathcal L\cong \mathcal O_X$$. There is an answer here on a pretty good website but I don't fully understand it. It defines non zero (why ?) morphisms $$\mathcal L\to \mathcal L$$ and $$\mathcal O_X\to \mathcal O_X$$ using composition of maps we get with two non zero global sections. Why would the regular sections defining these morphisms would be constant ? Why is an endomorphism of line bundles would an isomorphism ?

In my case we suppose more than in the answer, we have projective. With these assumptions is there another way ? I wanted to say that $$X$$ is a closed subscheme of some projective space but I think we would need more assumptions on $$S$$ if $$X=\operatorname{Proj}S$$.

Any help ?

Take a point $$x\in X$$ where the section of $$\mathcal L$$ and the section of $$\mathcal L^*$$ do not vanish. Since the fiber of a line bundle over $$x$$ is a 1-dimensional vector space, it is easy to see that compositions $$\mathcal L\to\mathcal L$$ and $$\mathcal O_X\to\mathcal O_X$$ induce non-zero endomorphisms of fibers over $$x$$, hence they are non-zero. Any endomorphism of line bundle is a multiplication with a regular function (one can prove it locally, using local triviality of a line bundle), and any regular function on a proper connected variety is constant.
• Perhaps another way to phrase this argument could be: let $s, s’$ be nonzero global sections of $\mathcal{L}$ and its dual, respectively. Then $f=s’(s)$ is a global function on $X$, hence is constant. If $f=0$, then $X=V(s) \cup V(s’)$ is not irreducible (since $s,s’$ are nonzero). Hence $f$ is a nonzero constant, so $s$ never vanishes. Sep 30 at 11:42
• @Aphelli Actually why does it imply $X=V(s)\cup V(s')$ ? If $x\notin V(s)$ I get that $s_x\in \operatorname{Ker}s'_x$, not necessarily that $s'_x=0$ Oct 10 at 9:51