The universal member away from the base locus

Local affine models. Let $V=\mathrm{Spec}A\subseteq X$ be an affine open on which $L$ has a generator $e$, and write $s_i{\mathrm{\mid }}_V=g_{i}e$ with $g_i\in A$; such charts exist because $L$ is invertible and $X$ is quasi-compact. For each chart $U_j=\mathrm{Spec}k[x_{\mathrm{\ell }}^{(j)}:\mathrm{\ell }\mathrm{\ne }j]$ of ${\mathbf{P}}_k^r$ put $I_{V,j}:=\mathrm{Spec}\text{\hspace{0.17em}⁣}(A[x_{\mathrm{\ell }}^{(j)}:\mathrm{\ell }\mathrm{\ne }j]/(g_j+{\sum }_{\mathrm{\ell }\mathrm{\ne }j}{g}_{\mathrm{\ell }}{x}_{\mathrm{\ell }}^{(j)}))\subseteq V{\times }_k{U}_j,$ the closed subscheme of the product cut out by the equation ${\sum }_i{\xi }_i{g}_i=0$ written in the chart $x_j^{(j)}=1$. This is a quotient presentation of a closed subscheme of the affine product [F5, F6], and the equation is the local equation of the general member in the sense of [F1] and [F2]. [F1, F2, F3, F5, F6, given, construct] 1.2 Independence of the choices. If $e^{\mathrm{\prime }}=ue$ is a second generator on $V$ with $u\in A^{\times }$ and $s_i{\mathrm{\mid }}_V=g_i^{\mathrm{\prime }}{e}^{\mathrm{\prime }}$, then $g_i^{\mathrm{\prime }}=u^{-1}{g}_i$, so the ideal $(g_j^{\mathrm{\prime }}+{\sum }_{\mathrm{\ell }\mathrm{\ne }j}{g}_{\mathrm{\ell }}^{\mathrm{\prime }}{x}_{\mathrm{\ell }})=u^{-1}(g_j+{\sum }_{\mathrm{\ell }\mathrm{\ne }j}{g}_{\mathrm{\ell }}{x}_{\mathrm{\ell }})$ is unchanged, and the two closed subschemes of $V{\times }_k{U}_j$ coincide. On the overlap $U_j\cap U_{j^{\mathrm{\prime }}}$ of the two standard charts, the transition formulas of [F3] identify the coordinates $x_{\mathrm{\ell }}^{(j)}={\xi }_{\mathrm{\ell }}\mathrm{/}{\xi }_j$ and $x_{\mathrm{\ell }}^{(j^{\mathrm{\prime }})}={\xi }_{\mathrm{\ell }}\mathrm{/}{\xi }_{j^{\mathrm{\prime }}}$ for the same homogeneous coordinates $\xi$, so substituting $x_{\mathrm{\ell }}^{(j)}=x_{\mathrm{\ell }}^{(j^{\mathrm{\prime }})}\mathrm{/}{x}_j^{(j^{\mathrm{\prime }})}$ and $x_{j^{\mathrm{\prime }}}^{(j)}=1\mathrm{/}{x}_j^{(j^{\mathrm{\prime }})}$ into the first equation and multiplying by the unit $x_j^{(j^{\mathrm{\prime }})}$ gives exactly the second equation $g_{j^{\mathrm{\prime }}}+{\sum }_{\mathrm{\ell }\mathrm{\ne }{j}^{\mathrm{\prime }}}{g}_{\mathrm{\ell }}{x}_{\mathrm{\ell }}^{(j^{\mathrm{\prime }})}=x_j^{(j^{\mathrm{\prime }})}(g_j+{\sum }_{\mathrm{\ell }\mathrm{\ne }j}{g}_{\mathrm{\ell }}{x}_{\mathrm{\ell }}^{(j)})\mathrm{.}$ Hence the local models agree on all overlaps of base charts and projective charts. [F1, F2, F3, algebra] 2.1 Gluing. The affine schemes $I_{V,j}$, indexed by a finite trivializing affine cover of $X$ and by $j=0,\dots ,r$, have pairwise compatible open immersions on their overlaps by step 1.2, so they glue along the principal opens of [F4] to a $k$-scheme $I_X$ together with a closed immersion $I_X\hookrightarrow X{\times }_k{\mathbf{P}}_k^r$ commuting with the two projections ${\pi }_X:I_X\to X$ and $p:I_X\to {\mathbf{P}}_k^r$. The scheme $I_X$ is finite type over $k$ because it is covered by the finitely many affine charts $I_{V,j}$, each a quotient of a finitely generated polynomial algebra over $k$. Define the incidence of the linear system to be the open subscheme $I:=I_X{\times }_X{X}^{\circ }$ obtained by base change along the open immersion $X^{\circ }\hookrightarrow X$, and keep $\pi :I\to X^{\circ }$ and $p:I\to {\mathbf{P}}_k^r$ for the restrictions. [F3, F4, F5, F6, step 1.1, step 1.2, algebra] 3.1 Local product structure. Fix $j$ and let $V=\mathrm{Spec}A\subseteq X_j^{\circ }$ be an affine chart trivializing $L$ by $e$, with $s_i{\mathrm{\mid }}_V=g_{i}e$. Since $V\cap \mathrm{\mid }Z(s_j)\mathrm{\mid }=\mathrm{\varnothing }$ and $Z(s_j)\cap V$ is cut out by $g_j$ by [F2], the function $g_j$ lies in no maximal ideal of $A$, hence $g_j\in A^{\times }$. Assume $r\ge 1$. For an incidence point $[{\xi }_0:\cdots :{\xi }_r]$ over $V$, the equation ${\sum }_i{g}_i{\xi }_i=0$ and invertibility of $g_j$ imply that some ${\xi }_{\mathrm{\ell }}$ with $\mathrm{\ell }\mathrm{\ne }j$ is nonzero: otherwise also ${\xi }_j=0$. Thus projection to the other coordinates defines an everywhere-defined map to ${\mathbf{P}}_k^{r-1}$, and its inverse is $[{\eta }_{\mathrm{\ell }}{]}_{\mathrm{\ell }\mathrm{\ne }j}⟼[{\xi }_j=-g_j^{-1}{\sum }_{\mathrm{\ell }\mathrm{\ne }j}{g}_{\mathrm{\ell }}{\eta }_{\mathrm{\ell }}:{\xi }_{\mathrm{\ell }}={\eta }_{\mathrm{\ell }}(\mathrm{\ell }\mathrm{\ne }j)]\mathrm{.}$ To check scheme morphisms rather than only point maps, choose ${\mathrm{\ell }}_0\mathrm{\ne }j$ and work on the chart ${\eta }_{{\mathrm{\ell }}_0}\mathrm{\ne }0$, equivalently the source chart ${\xi }_{{\mathrm{\ell }}_0}\mathrm{\ne }0$. Normalize ${\xi }_{{\mathrm{\ell }}_0}=1$. Its incidence ring is $A[y_m:m\mathrm{\ne }{\mathrm{\ell }}_0]/(g_{{\mathrm{\ell }}_0}+g_j{y}_j+{\sum }_{m\mathrm{\ne }j,{\mathrm{\ell }}_0}{g}_m{y}_m)\cong A[y_m:m\mathrm{\ne }j,{\mathrm{\ell }}_0],$ where the isomorphism eliminates $y_j$ by $y_j=-g_j^{-1}(g_{{\mathrm{\ell }}_0}+{\sum }_{m\mathrm{\ne }j,{\mathrm{\ell }}_0}{g}_m{y}_m)$. These charts cover the incidence over $V$, and their maps agree on overlaps because the displayed homogeneous formulas are scale invariant. They glue to ${\pi }^{-1}(V)\cong V{\times }_k{\mathbf{P}}_k^{r-1}$ over $V$, and the identifications agree under a change of trivialization of $L$ since all $g_i$ acquire the same unit factor. The affine $V$ cover $X_j^{\circ }$, giving the asserted product over $X_j^{\circ }$. In particular, the chart ${\xi }_j\mathrm{\ne }0$ alone need not cover the incidence; the eliminated coordinate is ${\xi }_j$, while the covering charts have ${\xi }_{{\mathrm{\ell }}_0}\mathrm{\ne }0$ for ${\mathrm{\ell }}_0\mathrm{\ne }j$. [F2, F3, F5, step 1.1, step 1.2, step 2.1, algebra] 3.2 Fibres. Let $[s]\in \mathbf{P}(W)(k)$, choose $0\mathrm{\ne }s\in W$ representing it and a $k$-basis $s_0,\dots ,s_r$ with $s=s_0$. Over an affine chart $V=\mathrm{Spec}A\subseteq X^{\circ }$ trivializing $L$ by $e$ with $s_i{\mathrm{\mid }}_V=g_{i}e$, the point $[s]$ lies in the chart $U_0$ of ${\mathbf{P}}_k^r$; the fibre of $p$ over it is obtained by substituting the coordinates $(1,0,\dots ,0)$ into $g_0+{\sum }_{\mathrm{\ell }\mathrm{\ne }0}{g}_{\mathrm{\ell }}{x}_{\mathrm{\ell }}^{(0)}$, that is, it is $\mathrm{Spec}(A\mathrm{/}(g_0))=Z(s_0)\cap V$ by [F2]. Since the trivializing charts cover $X^{\circ }$, this identifies the fibre with $Z(s)\cap X^{\circ }$. [F1, F2, F3, step 1.1, step 2.1, algebra] 4.1 Smoothness. Let $x\in I$. By step 3.1 some open neighbourhood of $x$ in $I$ is isomorphic over a smooth open subscheme of $X^{\circ }$ to $U{\times }_k{\mathbf{P}}_k^{r-1}$ when $r\ge 1$, while for $r=0$ the incidence is empty and the claim is vacuous. The open subscheme $X_j^{\circ }$ of the smooth $X$ is smooth over $k$ because smoothness is local on the source [F7]. The projective space ${\mathbf{P}}_k^{r-1}$ is smooth over $k$: its standard charts are polynomial algebras $k[x_1,\dots ,x_{r-1}]$, which are standard smooth with $c=0$ by [F8], and smoothness is local on the source [F7]. By [F9] each product $X_j^{\circ }{\times }_k{\mathbf{P}}_k^{r-1}$ is smooth over $k$, and these open pieces cover $I$, so $I\to \mathrm{Spec}k$ is smooth by [F7]. The Axiom of Choice enters only through the declared suppliers, notably [F9]; the finitely many charts and basis elements chosen here are finite choices and need no choice principle. [F7, F8, F9, F10, step 2.1, step 3.1, algebra] 5.1 The case $r=0$, and the boundary dispositions. If $r=0$ then $W=k{s}_0$ for $0\mathrm{\ne }{s}_0\in W$, so $\mathrm{Bs}(W)=\mathrm{\mid }Z(s_0)\mathrm{\mid }$ and $X^{\circ }=X\setminus \mathrm{\mid }Z(s_0)\mathrm{\mid }$. On every trivializing affine chart $V\subseteq X^{\circ }$ the coefficient $g_0$ is a unit by the argument of step 3.1, so the equation $g_0=0$ cuts out the empty subscheme; as the charts cover $X^{\circ }$, indeed $I=\mathrm{\varnothing }$, and the structural claims are vacuous. If $X=\mathrm{\varnothing }$ then $\mathrm{\Gamma }(X,L)=0$, so the hypothesis $W\mathrm{\ne }0$ has no instance. If $X^{\circ }=\mathrm{\varnothing }$, the incidence $I$ is empty by its definition, and the local product and fibre clauses are vacuous. The construction is canonical: a change of $k$-basis of $W$ multiplies the vector of coefficients $({\xi }_i)$ by an invertible constant matrix and hence induces an automorphism of ${\mathbf{P}}_k^r$ carrying the equation ${\sum }_i{\xi }_i{g}_i=0$ to itself, and a change of trivialization multiplies all $g_i$ by a unit; so $I$ is determined up to unique isomorphism compatible with both $\pi$ and $p$, equivalently over $X^{\circ }{\times }_k\mathbf{P}(W)$. Uniqueness follows because the glued quotient maps define a closed immersion into that product: a morphism over the product must be the identity on each quotient chart. This completes the proof.