The map defined by a base-point-free linear system

On $\mathbb{P}(V^{\ast })$, let $\gamma$ be the tautological line bundle whose fiber at a line $\mathrm{\ell }$ is $\mathrm{\ell }$. On the standard chart $U_i=\{Z_i\mathrm{\ne }0\}$ it has frame $v_i=(Z_0\mathrm{/}{Z}_i,\dots ,1,\dots ,Z_N\mathrm{/}{Z}_i)$. On $U_i\cap U_k$, $v_k=(Z_i\mathrm{/}{Z}_k)v_i$, so the dual frames ${ϵ}_i$ of ${\gamma }^{\ast }$ obey ${ϵ}_k=(Z_k\mathrm{/}{Z}_i){ϵ}_i$. These are holomorphic nowhere-zero transitions on the finite standard chart cover; [F3] constructs ${\gamma }^{\ast }$ as a holomorphic line bundle. The coordinate functional $Z_j$ restricted to each tautological line is a global holomorphic section of ${\gamma }^{\ast }$, with coefficient $Z_j\mathrm{/}{Z}_i$ in frame ${ϵ}_i$. Set ${\mathcal{O}}_{\mathbb{P}(V^{\ast })}(1):={\gamma }^{\ast }$; this is the line-bundle convention for projective space parametrizing lines.

For each $p$, base-point-freeness makes ${\mathrm{ev}}_p:V\to E_p$ a nonzero map to a one-dimensional space, hence surjective; its dual is injective and has one-dimensional image in $V^{\ast }$. This defines ${\phi }_V(p)$. In a local frame $e$ with $s_j=f_{j}e$, the vector ${\mathrm{ev}}_p^{\ast }(e_p^{\ast })$ has coordinates $(f_0(p),\dots ,f_N(p))$ in the dual basis, so at least one coordinate is nonzero and the projective expression is $[f_0(p):\cdots :f_N(p)]$. Replacing $e$ by $ue$ for a nowhere-zero holomorphic $u$ multiplies every $f_j$ by $u^{-1}$, leaving this projective line unchanged.

If $s_k^{\mathrm{\prime }}={\sum }_j{a}_{kj}{s}_j$, then in every local frame $f_k^{\mathrm{\prime }}={\sum }_j{a}_{kj}{f}_j$, so the coordinate vector changes by $A$ and ${\phi }_{B^{\mathrm{\prime }}}=\mathbb{P}(A)\circ {\phi }_B$. The intrinsic map to $\mathbb{P}(V^{\ast })$ was defined from evaluation and is independent of any basis. Also, by [F2], every nonzero $h\in V\subseteq L(D)$ gives the effective divisor $(h)+D$; this set is defined by the subspace $V$ itself, so $\mathrm{\mid }V\mathrm{\mid }$ is basis-independent. The coordinate image can move: on $X={\mathbb{P}}^1$, $D=2[\mathrm{\infty }]$, and $V=⟨1,z,z^2⟩$, all three polynomials have pole order at most $2$ at infinity and no poles elsewhere, so they lie in $L(D)$ and are linearly independent. At every finite point the section $1s_D$ is nonzero, and at infinity $z^2{s}_D$ is nonzero because $(z^2)+D=2[0]$; hence $V$ is base-point-free. The two bases $(1,z,z^2)$ and $(1,z,z^2+1)$ give images satisfying $XZ=Y^2$ and $XZ-X^2=Y^2$, respectively; $[1:0:0]$ lies in the first image and not the second, while the bases are related by $[X:Y:Z]\mapsto [X:Y:Z+X]$.

The open sets $X_k=\{p:s_k(p)\mathrm{\ne }0\}$ cover $X$. On $X_k$ the image lies in $U_k$, and its target chart coordinates are $f_j\mathrm{/}{f}_k$ for $j\mathrm{\ne }k$, which are holomorphic because $f_k$ is nowhere zero there. These local expressions are continuous and holomorphic, agree on overlaps by the common-factor calculation in step 1.2, and therefore define the unique canonical holomorphic map.

At $p$, the tautological fiber ${\gamma }_{{\phi }_V(p)}$ is $\mathrm{im}({\mathrm{ev}}_p^{\ast })$. The canonical evaluation pairing defines ${\mathrm{\Psi }}_p:E_p\to {\gamma }_{{\phi }_V(p)}^{\ast }$ by ${\mathrm{\Psi }}_p(v)({\mathrm{ev}}_p^{\ast }\lambda )=\lambda (v)$ for $\lambda \in E_p^{\ast }$. Since ${\mathrm{ev}}_p$ is surjective, this is a linear isomorphism. On $X_k$, the local formula is $\mathrm{\Psi }(e)=f_k^{-1}{\phi }_V^{\ast }{ϵ}_k$. It agrees on different frames and charts because ${ϵ}_l=(Z_l\mathrm{/}{Z}_k){ϵ}_k$ and $Z_l\mathrm{/}{Z}_k\circ {\phi }_V=f_l\mathrm{/}{f}_k$; hence the fiberwise isomorphisms form a holomorphic bundle isomorphism. Moreover $\mathrm{\Psi }(s_j)=f_j{f}_k^{-1}{\phi }_V^{\ast }{ϵ}_k={\phi }_V^{\ast }{Z}_j$. Thus the pullback identity and the coordinate-section claim hold, with no Choice principle used.