Relative very ampleness implies relative ampleness

Forms give sections with the same nonvanishing locus. Let $U=\mathrm{Spec}A$ be an affine open of $S$ and let $F\in A[x_0,\dots ,x_n]$ be homogeneous of degree $d>0$. On the chart $U_j=D_{+}(x_j)$ put $F^{(j)}=F(x_0^{(j)},\dots ,1,\dots ,x_n^{(j)})=F\mathrm{/}{x}_j^d$, and define $s_F{\mathrm{\mid }}_{U_j}=F^{(j)}{e}_j^d\in \mathrm{\Gamma }(U_j,\mathcal{O}(d))$. On an overlap the coordinates satisfy $x_{\mathrm{\ell }}^{(j)}=x_{\mathrm{\ell }}^{(i)}\mathrm{/}{x}_j^{(i)}$, so $F^{(j)}=F^{(i)}\mathrm{/}(x_j^{(i)}{)}^d$, and the frame transition $e_j^d=(x_j^{(i)}{)}^d{e}_i^d$ gives $F^{(j)}{e}_j^d=F^{(i)}{e}_i^d$; hence the local sections glue to a global section $s_F\in \mathrm{\Gamma }({\mathbb{P}}_U^n,\mathcal{O}(d))$. Since each $e_j$ is a frame, the nonvanishing locus is computed on charts as $X_{s_F}\cap U_j=\{F^{(j)}\mathrm{\ne }0\}$, so $X_{s_F}=D_{+}(F)$.

The restricted situation. Put $X_U=f^{-1}(U)$ with structure morphism $f_U:X_U\to U$ and $L_U=L{\mathrm{\mid }}_{X_U}$. By [F2] the morphism $f$ is quasi-compact, so $X_U$ is quasi-compact, and the base change $i_U:X_U\to {\mathbb{P}}_U^n$ of $i$ along $U\hookrightarrow S$ is a quasi-compact immersion with $L_U\cong i_U^{\ast }{\mathcal{O}}_{{\mathbb{P}}_U^n}(1)$; this is the situation of [F1] over the affine base $U$.

Shrinking a neighbourhood to a standard open. Let $x\in X_U$ and let $W\subseteq X_U$ be an affine open subscheme containing $x$; write $y=i_U(x)$. Since $i_U$ is an immersion, it is a homeomorphism onto the locally closed subset $i_U(X_U)\subseteq {\mathbb{P}}_U^n$, so $i_U(W)$ is open in $i_U(X_U)$ and there is an open $V\subseteq {\mathbb{P}}_U^n$ with $i_U^{-1}(V)\subseteq W$ and $y\in V$. By [F3] the standard opens $D_{+}(F)$ with $F$ homogeneous of positive degree form a basis of the topology, so choose such an $F$ with $y\in D_{+}(F)\subseteq V$. Then $x\in i_U^{-1}(D_{+}(F))\subseteq i_U^{-1}(V)\subseteq W$.

Pulling back the sections. For $F$ homogeneous of positive degree $d$, the pullback $i_U^{\ast }{s}_F$ is a global section of $i_U^{\ast }\mathcal{O}(d)\cong L_U^d$, and its nonvanishing locus is $X_{i_U^{\ast }{s}_F}=i_U^{-1}(X_{s_F})=i_U^{-1}(D_{+}(F))$: the pullback of a section of an invertible sheaf has nonvanishing locus the preimage of the original nonvanishing locus, because a local trivialisation of $\mathcal{O}(d)$ pulls back to one of $L_U^d$ and the corresponding function is the pullback function.

Ampleness at a point. With $F$ as in step 1.3 and $d=\mathrm{deg}F>0$, let $s=i_U^{\ast }{s}_F\in \mathrm{\Gamma }(X_U,L_U^d)$, a section of a positive power of the invertible sheaf $L_U$. Then $x\in X_s=i_U^{-1}(D_{+}(F))\subseteq W$ by step 2.1, and since $W$ is an affine open subscheme of $X_U$, [F4] gives that $X_s=W\cap X_s$ is an affine open subscheme of $X_U$. So every point of $X_U$ admits a positive power of $L_U$ with a global section whose nonvanishing locus is affine and contains the point.

Ampleness over an affine base open. The scheme $X_U$ is quasi-compact by step 1.2, so the criterion [F5] applies to the invertible sheaf $L_U$ on $X_U$ with the sections produced in step 3.1: $L_U$ is ample on $X_U$.

Conclusion. Every affine open $U\subseteq S$ has $L{\mathrm{\mid }}_{f^{-1}(U)}$ ample on $f^{-1}(U)$, so $L$ is $f$-ample by definition; if $S=\mathrm{Spec}R$ is affine this is absolute ampleness of $L$ on $X$. If in addition the immersion $i$ is a closed immersion, the same argument applies verbatim; the only simplification in that case is that the image is closed, so the shrinking step 1.3 may be replaced by choosing a chart $U_i$ containing $i(x)$, whose preimage $X\cap U_i$ is affine as a closed subscheme of the affine scheme $U_i$. If $X=\mathrm{\varnothing }$ or $S=\mathrm{\varnothing }$ there is no point to test and the conditions of [F5] and of $f$-ampleness are vacuous, so the conclusion holds. The Axiom of Choice [A1] is inherited from the Proj and associated-sheaf constructions; no choice is made here. [A1, F1, F5, step 4.1, cases: empty and affine base] \qed