Generating line-bundle sections define a morphism to projective space

The charts of the construction. By [F1] the open sets $X_{s_i}$ cover $X$. By [F2] each $s_i$ trivialises $L$ over $X_{s_i}$, so for every $i$ and every $j$ the ratio $s_j\mathrm{/}{s}_i$ is a regular function on $X_{s_i}$: $s_j\mathrm{/}{s}_i=(s_j\otimes s_i^{-1}){\mathrm{\mid }}_{X_{s_i}}$ under the identification $L\otimes L^{-1}={\mathcal{O}}_X$, restricted to $X_{s_i}$.

The local morphisms. Fix $i$ and let $U_i\subseteq {\mathbb{P}}_S^n$ be the $i$-th chart, an affine space over $S$ with coordinates $x_j^{(i)}$, $j\mathrm{\ne }i$, by [F3]. By [F4] the $n$ regular functions $s_j\mathrm{/}{s}_i$, $j\mathrm{\ne }i$, on $X_{s_i}$ define a morphism ${\phi }_i:X_{s_i}\to U_i$ over $S$ with $x_j^{(i)}\circ {\phi }_i=s_j\mathrm{/}{s}_i$ for every $j\mathrm{\ne }i$; this is the unique $S$-morphism with these coordinates. Its image lies in $U_i$, so ${\phi }_i^{-1}(D_{+}(x_i))=X_{s_i}$; moreover on $X_{s_i{s}_j}$ the quotient $s_j\mathrm{/}{s}_i$ is invertible by [F2], so ${\phi }_i(X_{s_i{s}_j})\subseteq D(x_j^{(i)})\subseteq U_i$.

Compatibility on overlaps. On $X_{s_i}\cap X_{s_j}=X_{s_i{s}_j}$ both quotients $s_j\mathrm{/}{s}_i$ and $s_i\mathrm{/}{s}_j$ are invertible, and for $\mathrm{\ell }\mathrm{\ne }i,j$ the transition formula of [F3] reads $x_{\mathrm{\ell }}^{(j)}=x_{\mathrm{\ell }}^{(i)}\mathrm{/}{x}_j^{(i)}$; substituting $x_{\mathrm{\ell }}^{(i)}=s_{\mathrm{\ell }}\mathrm{/}{s}_i$ and $x_j^{(i)}=s_j\mathrm{/}{s}_i$ from step 1.2 gives $s_{\mathrm{\ell }}\mathrm{/}{s}_j$, which is $x_{\mathrm{\ell }}^{(j)}\circ {\phi }_j$; likewise $x_i^{(j)}=1\mathrm{/}{x}_j^{(i)}$ corresponds to $s_i\mathrm{/}{s}_j$. Hence ${\phi }_i$ and ${\phi }_j$ agree on the overlap.

The global morphism. The local morphisms ${\phi }_i$ of step 1.2 are compatible on overlaps by step 2.1, and the opens $X_{s_i}$ cover $X$ by step 1.1; hence they glue to a morphism $\phi :X\to {\mathbb{P}}_S^n$ by [F5], which is a morphism over $S$ because each ${\phi }_i$ is. Its restrictions satisfy ${\phi }^{-1}(D_{+}(x_i))\cap X_{s_i}=X_{s_i}\cap X_{s_i}=X_{s_i}$ and, on $X_{s_j}$, ${\phi }^{-1}(D_{+}(x_i))=X_{s_i{s}_j}$, so altogether ${\phi }^{-1}(D_{+}(x_i))=X_{s_i}$.

The pullback of the twist. On $X_{s_i}$ define an isomorphism ${\phi }^{\ast }\mathcal{O}(1){\mathrm{\mid }}_{X_{s_i}}\to L{\mathrm{\mid }}_{X_{s_i}}$ by sending the pullback of the frame $e_i$ to $s_i$; this is an isomorphism of invertible sheaves because both sides are free of rank one there. On the overlap $X_{s_i{s}_j}$ the transition $e_i=x_i^{(j)}{e}_j$ of [F3] pulls back to the scalar $x_i^{(j)}\circ \phi =s_i\mathrm{/}{s}_j$: one has ${\phi }^{\ast }(e_i)={\phi }^{\ast }(x_i^{(j)}{e}_j)=(s_i\mathrm{/}{s}_j){\phi }^{\ast }(e_j)$ with ${\phi }^{\ast }(e_j)=s_j$, so ${\phi }^{\ast }(e_i)$ corresponds to $s_i$ under the trivialisation on $X_{s_j}$ exactly as it does under the trivialisation on $X_{s_i}$; hence the local isomorphisms glue to an isomorphism ${\phi }^{\ast }\mathcal{O}(1)\cong L$ under which ${\phi }^{\ast }(e_i)$ corresponds to $s_i$, that is, the universal coordinate section $x_i$ pulls back to $s_i$.

Uniqueness. Let $\psi :X\to {\mathbb{P}}_S^n$ be an $S$-morphism with ${\psi }^{\ast }\mathcal{O}(1)\cong L$ carrying the coordinate sections to $s_0,\dots ,s_n$. Then ${\psi }^{-1}(D_{+}(x_i))=X_{s_i}$: a point maps into $D_{+}(x_i)$ exactly when the pullback of the coordinate $x_i$ does not vanish there, and that pullback is $s_i$. On $X_{s_i}$ the coordinates satisfy $x_j^{(i)}\circ \psi ={\psi }^{\ast }(x_j)\mathrm{/}{\psi }^{\ast }(x_i)=s_j\mathrm{/}{s}_i$, which agrees with $\phi$ by step 2.1; hence $\psi =\phi$ on each $X_{s_i}$, and since these cover $X$, $\psi =\phi$ by [F5].

Conclusion. Steps 1.1 to 3.1 construct the $S$-morphism with ${\phi }^{-1}(D_{+}(x_i))=X_{s_i}$, step 4.1 identifies ${\phi }^{\ast }\mathcal{O}(1)$ with $L$ compatibly with the given sections, and step 5.1 proves uniqueness. Nothing in the construction asserts injectivity or immersion: two distinct points may have proportional tuples, and $\phi$ is an immersion only under additional hypotheses. If some $s_i=0$ then $X_{s_i}=\mathrm{\varnothing }$ and the corresponding local piece is empty, which the gluing of step 3.1 allows; if all $s_i=0$ the hypothesis that they generate $L$ fails unless $X=\mathrm{\varnothing }$, and then the construction is vacuous. The Axiom of Choice [A1] is inherited from the projective-space and associated-sheaf constructions; no choice is made here. [A1, step 3.1, step 4.1, step 5.1, cases: vanishing sections and empty X] \qed