A base-point-free linear system defines a morphism to projective space

The generating data. By [F10], $L(D)$ and hence its subspace $V$ are finite-dimensional; under the Axiom of Choice choose a $k$-basis $f_0,\dots ,f_r$, where $r+1={\dim }_{k}V\ge 1$. Let $s_i\in \mathrm{\Gamma }(C,{\mathcal{O}}_C(D))$ be the section corresponding to $f_i$ under the dictionary of [F2]. Since $V$ is base-point-free, [F1] says that the evaluation morphism ${\mathrm{ev}}_V:{\mathcal{O}}_C^{r+1}\to {\mathcal{O}}_C(D)$, $(g_i)\mapsto {\sum }_i{g}_i{s}_i$, is surjective; equivalently the global sections $s_0,\dots ,s_r$ generate ${\mathcal{O}}_C(D)$ in the sense of Global generation by the evaluation map.

The converse: data of a morphism. Let $\phi :C\to {\mathbb{P}}_k^r$ be a $k$-morphism and let $\alpha :{\phi }^{\ast }\mathcal{O}(1)\to {\mathcal{O}}_C(D)$ be an isomorphism; put $s_i=\alpha ({\phi }^{\ast }{x}_i)\in \mathrm{\Gamma }(C,{\mathcal{O}}_C(D))$. Since the coordinate sections $x_0,\dots ,x_r$ generate $\mathcal{O}(1)$ by [F5] and pullback and $\alpha$ are isomorphisms of invertible sheaves, the sections $s_0,\dots ,s_r$ generate ${\mathcal{O}}_C(D)$; in particular they are not all zero. Let $f_0,\dots ,f_r\in L(D)$ be the rational functions corresponding to $s_0,\dots ,s_r$ under [F2] and let $V_{\phi }\subseteq L(D)$ be their $k$-span. By [F9], the coordinate sections span $H^0({\mathbb{P}}_k^r,\mathcal{O}(1))$, so $V_{\phi }$ is exactly the image of the pullback map on global sections followed by $\alpha$ and the section dictionary. It is base-point-free because the sections $s_i$ generate ${\mathcal{O}}_C(D)$ (equivalently, by [F1], because the evaluation morphism of the data is surjective). By [F4] the morphism attached to the generating data $({\mathcal{O}}_C(D);s_0,\dots ,s_r)$ by the universal property of [F3] is $\phi$ itself, since these data are the image under $\alpha$ of the data of $\phi$.

The morphism. Apply [F3] with the base $S=\mathrm{Spec}k$, the source $X=C$, the invertible sheaf $L={\mathcal{O}}_C(D)$ and the generating sections $s_0,\dots ,s_r$: there is a unique $k$-morphism ${\phi }_V:C\to {\mathbb{P}}_k^r$ with an isomorphism ${\mathcal{O}}_C(D)\cong {\phi }_V^{\ast }\mathcal{O}(1)$ carrying the coordinate section $x_i$ to $s_i$, and with ${\phi }_V^{-1}(D_{+}(x_i))=C_{s_i}$ the locus where $s_i$ is nonvanishing. This is the first assertion of the statement.

Independence of the isomorphism. If ${\alpha }^{\mathrm{\prime }}$ is another isomorphism ${\phi }^{\ast }\mathcal{O}(1)\to {\mathcal{O}}_C(D)$, then ${\alpha }^{\mathrm{\prime }}=\alpha \circ \theta$ for an automorphism $\theta$ of ${\phi }^{\ast }\mathcal{O}(1)$, and $\theta$ is multiplication by a global unit of $C$, that is, by an element $c\in k^{\times }$ by [F7]; scalars act on the whole space of sections, so the image subspace $V_{\phi }$ and the generating data up to isomorphism are unchanged.

Hyperplanes pull back to members of the system. Let $\lambda =({\lambda }_0,\dots ,{\lambda }_r)\mathrm{\ne }0$ be a $k$-tuple, let $H_{\lambda }$ be the scheme-theoretic zero divisor of the corresponding linear form on ${\mathbb{P}}_k^r$ (empty when $r=0$), and let $f_{\lambda }={\sum }_i{\lambda }_i{f}_i\in V\setminus \{0\}$; the latter is nonzero because $f_0,\dots ,f_r$ is a basis. By [F5] the local equations of $H_{\lambda }$ define an effective Cartier divisor, including the empty divisor for $r=0$. The section $s_{f_{\lambda }}$ is nonzero and has nonzero generic value by [F2]. Since $C$ is integral, each local coefficient of this section in a frame is a nonzero element of a domain, hence a nonzerodivisor. Thus the scheme-theoretic pullback of $H_{\lambda }$ is an effective Cartier divisor: its ideal is locally generated by the pulled-back equations, equivalently by the pullback section. Under the isomorphism of step 2.1 this section is ${\phi }_V^{\ast }({\sum }_i{\lambda }_i{x}_i)={\sum }_i{\lambda }_i{s}_i=s_{f_{\lambda }}$. Its Cartier divisor is the rational-section divisor of $s_{f_{\lambda }}$; under [F2]'s Cartier/Weil identification the associated Weil divisor is $\mathrm{div}(f_{\lambda })+D$, with vanishing multiplicities included.

Independence of the basis. Suppose $g_0,\dots ,g_r$ is another $k$-basis of $V$, with $g_j={\sum }_i{a}_{ij}{f}_i$ for an invertible matrix $A=(a_{ij})\in {\mathrm{G}\mathrm{L}}_{r+1}(k)$, and let ${\tau }_A$ be the automorphism of ${\mathbb{P}}_k^r$ induced by $A$ on coordinates. The universal property [F3] applied to the basis $g$ produces the unique morphism ${\phi }^{\mathrm{\prime }}$ with ${\phi }^{\mathrm{\prime }\ast }(x_j)\mapsto s_{g_j}={\sum }_i{a}_{ij}{s}_{f_i}$; since ${\phi }_V^{\ast }(x_j\circ {\tau }_A)={\phi }_V^{\ast }({\sum }_i{a}_{ij}{x}_i)={\sum }_i{a}_{ij}{s}_{f_i}$, the morphism ${\tau }_A\circ {\phi }_V$ has that same property, so by uniqueness ${\phi }^{\mathrm{\prime }}={\tau }_A\circ {\phi }_V$. Thus the morphism depends on $V$ only up to composition with the standard projective-linear action of ${\mathrm{P}\mathrm{G}\mathrm{L}}_{r+1}(k)$ on the target, as asserted.

The members of $P(V)$ are the hyperplane pullbacks. Every nonzero $f\in V$ is ${\sum }_i{\lambda }_i{f}_i$ for a unique projective tuple $[\lambda ]\in P(V)$, and every nonzero $\lambda$ arises; conversely a scalar multiple of $\lambda$ changes $f_{\lambda }$ by a scalar and leaves both $H_{\lambda }$ and $\mathrm{div}(f_{\lambda })+D$ unchanged. Hence the assignment sending $[\lambda ]$ to the Weil divisor associated to the scheme-theoretic Cartier pullback ${\phi }_V^{\ast }{H}_{\lambda }$ is a well-defined bijection onto $P(V)=\{\mathrm{div}(f)+D:f\in V\setminus \{0\}\}$. For $r=0$, both sets are singletons: the only such hyperplane is empty and the only member of $P(V)$ is the zero divisor.

The nondegenerate case. Suppose in addition that ${\phi }^{\ast }{x}_0,\dots ,{\phi }^{\ast }{x}_r$ are linearly independent in $\mathrm{\Gamma }(C,{\phi }^{\ast }\mathcal{O}(1))$; equivalently, since $\alpha$ is injective on sections, that $s_0,\dots ,s_r$ are linearly independent, equivalently ${\dim }_k{V}_{\phi }=r+1$. Then $V_{\phi }$ is a base-point-free subspace of $L(D)$ of dimension $r+1$, and by step 1.1 and step 2.1 the morphism ${\phi }_{V_{\phi }}$ attached to the ordered basis $f_0,\dots ,f_r$ of $V_{\phi }$ is the morphism attached to the data $(s_0,\dots ,s_r)$, hence ${\phi }_{V_{\phi }}=\phi$; by step 3.2 any other choice of basis changes this morphism by the standard projective-linear action. This is the converse of the statement.

Conclusion. Steps 1.1 to 1.2 construct the morphism ${\phi }_V$ together with the isomorphism to ${\phi }_V^{\ast }\mathcal{O}(1)$, step 4.1 identifies the divisors of $P(V)$ with the pullbacks of hyperplanes, step 3.2 records the dependence on the basis, steps 1.2, 2.1, 3.2 and 4.2 give the converse with the independence of the section dictionary in the isomorphism, and the final clause of the statement is exactly the last sentence of [F1]. The Axiom of Choice is inherited from proper coherent cohomology [F10] and the projective-space constructions used in [F3] and [F4]; it also supplies the DC premise of the Cartier-to-Weil interface.