A degree-zero line bundle with a nonzero section is trivial

Given: the Axiom of Choice inherited from the degree, Cartier and structure-sheaf cohomology suppliers, with Dependent Choice supplied through AC implies DC implies countable choice; a field $k$, a smooth proper geometrically integral curve $C$ over $k$, an invertible sheaf $\mathcal{L}$ on $C$ with ${\mathrm{deg}}_k\mathcal{L}=0$, and the hypothesis $H^0(C,\mathcal{L})\mathrm{\ne }0$.

Cohomology and global sections: $H^0(C,\mathcal{L})=\mathrm{\Gamma }(C,\mathcal{L})$ is the group of global sections of the abelian sheaf $\mathcal{L}$, so a nonzero cohomology group gives a nonzero section $s\in \mathrm{\Gamma }(C,\mathcal{L})$; $\mathcal{L}$ is invertible, and $C$ is integral (Sheaf cohomology as right derived global sections, Global sections of an abelian sheaf, Invertible sheaves, Integral schemes). Such an $s$ has nonzero generic germ. Indeed, if $s_{\eta }=0$, the definition of a stalk gives a nonempty open $U$ on which $s$ vanishes. On any nonempty affine open $V=\mathrm{Spec}A$ trivializing $\mathcal{L}$, write $s{\mathrm{\mid }}_V=fe$ in a frame $e$. Since $C$ is integral, $A$ is a domain; since $C$ is irreducible, $U\cap V$ is a nonempty open and contains a nonempty distinguished open $D(a)$ for some $a\mathrm{\ne }0$. The coefficient $f$ becomes zero in $A_a$. The localization map $A\to A_a$ is injective because $A$ is a domain and $a\mathrm{\ne }0$, so $f=0$. Thus $s$ vanishes on every such $V$, hence globally, a contradiction.

The current Rational sections of line bundles are Cartier divisors says that a nonzero rational section $s$ of an invertible sheaf $\mathcal{L}$ determines a Cartier divisor ${\mathrm{div}}_C(s)$ with ${\mathcal{O}}_C({\mathrm{div}}_C(s))\cong \mathcal{L}$. By [F1] a nonzero global section has a nonzero generic germ, so this interface applies to it; its regularity makes the associated Cartier divisor effective. The current Invertible sheaf of cartier divisor constructs ${\mathcal{O}}_C(D)$ and gives ${\mathcal{O}}_C(0)\cong {\mathcal{O}}_C$. The degree homomorphism The degree of a divisor descends to the Picard group of a normal proper curve satisfies ${\mathrm{deg}}_k{\mathcal{O}}_C(D)={\mathrm{deg}}_{k}D$ for every divisor $D$ on the normal proper curve.

The current Cartier-to-Weil theorem identifies Cartier divisors on the smooth proper geometrically integral curve $C$ with the divisors of Divisors on a smooth proper curve, compatibly with principal divisors and local orders. Its local rings at closed points are DVRs and its generic local ring is a field, so $C$ is normal; the normal proper curve hypothesis of [F2] is therefore met. In particular, the effective Cartier divisor ${\mathrm{div}}_C(s)$ of [F2] gives an effective divisor on $C$ (Cartier and Weil divisors agree on a smooth curve, Divisors on a smooth proper curve, Principal weil divisor and class group).

Effective divisors: for an effective divisor $D={\sum }_x{n}_x[x]$ on a proper geometrically integral curve, ${\mathrm{deg}}_k(D)={\sum }_x{n}_x[\kappa (x):k]$ is a nonnegative integer, and ${\mathrm{deg}}_k(D)=0$ if and only if $D=0$ (Effective divisors have nonnegative degree).

The structure sheaf: the canonical map $k\to H^0(C,{\mathcal{O}}_C)$ is an isomorphism, so $H^0(C,{\mathcal{O}}_C)\cong k\mathrm{\ne }0$ and the structure sheaf has a nonzero global section; and ${\mathrm{deg}}_k{\mathcal{O}}_C={\mathrm{deg}}_k(0)=0$ under the dictionary of [F2] (Functions on a proper curve, Sheaf cohomology as right derived global sections).

The Axiom of Choice is inherited through the degree, Cartier-to-Weil and structure-sheaf cohomology suppliers of [F2], [F3] and [F5]. The Cartier-to-Weil supplier requires Dependent Choice, supplied from AC by AC implies DC implies countable choice. The proof selects one nonzero section of the given nonzero space and makes no further selection (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain).