Effective divisors linearly equivalent to D are sections modulo scalars

Given: A smooth proper geometrically integral curve $C$ over a field $k$ with function field $k(C)$, a divisor $D$ on $C$, and the space $L(D)$ of The space L(D); the Axiom of Choice is assumed.

A divisor on $C$ is a finite formal $\mathbb{Z}$-linear combination $D={\sum }_x{n}_x[x]$ of closed points, it is effective when all $n_x\ge 0$, and the divisor of a nonzero rational function $f$ is $\mathrm{div}(f)={\sum }_x{\mathrm{ord}}_x(f)[x]$, the order being the discrete valuation of the local ring ${\mathcal{O}}_{C,x}$, a discrete valuation ring; the order is additive, vanishes exactly on units, and ${\mathrm{ord}}_x(c)=0$ for $c\in k^{\times }$. (Divisors on a smooth proper curve, Divisor support positive negative parts, Order codimension one rational function, Local rings at closed points of smooth curves are discrete valuation rings, Curves over a field)

$L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}$ is a $k$-subspace of $k(C)$, and $f\in L(D)\setminus \{0\}$ means exactly that $\mathrm{div}(f)+D$ is an effective divisor. (The space L(D))

The current Cartier dictionary has these interfaces. The definition Linear equivalence cartier divisors says $D\sim D^{\mathrm{\prime }}$ exactly when $D-D^{\mathrm{\prime }}={\mathrm{div}}_C(u)$ for a global meromorphic unit $u$; Effective cartier divisor defines effectivity by local equations that are regular sections, meaning multiplication is injective on every stalk; and Invertible sheaf of cartier divisor defines ${\mathcal{O}}_C(D)$ by the local sheaves $f_i^{-1}{\mathcal{O}}_{U_i}$. The theorem Rational sections of line bundles are Cartier divisors associates to a nonzero rational section its Cartier divisor and an isomorphism from the sheaf of that divisor carrying the canonical section to the given section; A regular global section of an invertible sheaf glues to an effective Cartier divisor constructs an effective Cartier divisor from a regular global section. On $C$, Cartier and Weil divisors agree on a smooth curve identifies these Cartier conventions with the closed-point Weil-divisor conventions and preserves principal divisors. (Linear equivalence cartier divisors, Effective cartier divisor, Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors, A regular global section of an invertible sheaf glues to an effective Cartier divisor, Cartier and Weil divisors agree on a smooth curve)

In ZF, AC implies DC by AC implies DC implies countable choice. Under the stated AC assumption, the current Cartier and Weil divisors agree on a smooth curve body applies with its DC premise: on $C$, every divisor is Cartier, the Cartier and Weil divisor groups are identified, and the identification is compatible with principal divisors. Thus a Weil divisor difference $D^{\mathrm{\prime }}-D$ equals $\mathrm{div}(f)$ for a nonzero rational function $f$ exactly when the corresponding Cartier divisors are linearly equivalent in the sense of [F3]. (Cartier and Weil divisors agree on a smooth curve, AC implies DC implies countable choice, Divisors on a smooth proper curve)

Under AC, a proper curve over $k$ that is geometrically connected and geometrically reduced has $H^0(C,{\mathcal{O}}_C)=k$, with the map $k\to H^0(C,{\mathcal{O}}_C)$ an isomorphism (Functions on a proper curve). A smooth proper geometrically integral curve is such a curve; hence a rational function $h\in k(C{)}^{\times }$ with $\mathrm{div}(h)=0$ has no poles and lies in $H^0(C,{\mathcal{O}}_C)=k$, and $h\mathrm{\ne }0$ gives $h\in k^{\times }$. (Functions on a proper curve, The Axiom of Choice, Divisors on a smooth proper curve)