Projective weak models and rational mapping

By [F1] choose a projective embedding of $A$ over $K$, let $X$ be the schematic closure of $A$ in the corresponding projective $R$-space, and note that $X$ is proper of finite type over $R$ with generic fibre $A$. Its chart rings are $\pi$-torsion-free, since schematic closure contracts the generic ideal, so $X$ is flat over the DVR. For every $a\in A(K^{\mathrm{s}\mathrm{h}})=X(K^{\mathrm{s}\mathrm{h}})$, the valuative criterion of properness extends $a$ uniquely to an $R^{\mathrm{s}\mathrm{h}}$-point of $X$; applying the finite permissible smoothening theorem [F1] to $X$ produces a finite sequence of special-fibre blowups, proper and generically identical, whose smooth locus contains every such section. The smooth locus $V$ of the resulting model is therefore a smooth separated finite-type $R$-model of $A$ with $V(R^{\mathrm{s}\mathrm{h}})=A(K^{\mathrm{s}\mathrm{h}})$, proving (a).

For a fibre-dense open $U$ in a smooth scheme over a Noetherian base, apply the filtration argument of [F2] on an affine source chart. More explicitly, a function zero on $U$ maps to zero in the last factor's generic fibre, because that fibre is reduced and $U$ meets every irreducible component. Flatness over the domain $B\mathrm{/}\mathfrak{p}$ injects the factor into its generic fibre. The function therefore lies in the previous filtration submodule, where it still restricts to zero; induction through the finite filtration gives zero. Thus restriction is injective. The same proof applies after any Noetherian base change for which the source remains smooth and $U$ remains fibre-dense, in particular the DVR localizations and strict henselizations used here. Schematic graph closures commute with flat base change because restriction kernels do: compute restriction with a finite affine cover of $U$, use its finite equalizer, and tensor with a flat algebra. No claim that a smooth algebra over an arbitrary Noetherian base has only minimal associated primes is used.

The graph closure of a rational map is computed on a finite affine cover by kernels of restriction maps, so it commutes with flat base change and the domain of definition is the open where the graph projection is an isomorphism; if a faithfully flat pullback of that projection is an isomorphism, affine ring descent gives the isomorphism before pullback by [F3]. Hence domains descend along faithfully flat source maps and commute with flat base changes, and representatives agreeing on a schematically dense open of a separated target are equal. This is the BLR relative-rational-map statement of Chapter 2, Section 5 in the form used below.

For (b), shrink $Z$ fibre-densely so the generic rational map is defined on its whole generic fibre: the closure of the excluded proper generic closed set is nowhere dense in the smooth irreducible special fibre by the DVR dimension argument. Let ${\mathrm{\Gamma }}_i\subseteq Z{\times }_R{V}_i$ be its schematic graph closure for each of the finitely many members of the weak collection, with projection $p_i$ to $Z$. After passing to $R^{\mathrm{s}\mathrm{h}}$, every rational special point of $Z$ lifts to a section by [F1]; its generic image extends to a section of some $V_i$ by weakness, and the paired section lies in ${\mathrm{\Gamma }}_i$ by schematic closure. These special points are dense. Constructibility [F3] and finiteness of the collection therefore force some image $p_i({\mathrm{\Gamma }}_i)$ to contain the generic point $\eta$ of the original special fibre (this can be checked after the faithfully flat strict-henselian extension). Choose $q\in {\mathrm{\Gamma }}_i$ over $\eta$. The graph is flat and generically isomorphic to $Z$, so ${\mathcal{O}}_{{\mathrm{\Gamma }}_i,q}$ and the DVR ${\mathcal{O}}_{Z,\eta }$ lie in the same function field. The former is a local ring dominating that DVR; a proper overring of a DVR in its fraction field inverts the uniformizer and cannot dominate it, so the two rings coincide. Finite-type affine presentations and clearing denominators spread this stalk equality to an isomorphism on neighbourhoods of $q$ and $\eta$. Composing its inverse with the other graph projection extends the generic map on an $R$-dense open of $Z$, proving (b).

For (c), put $R^{\mathrm{\prime }}={\mathcal{O}}_{Z,\eta }$ and $K^{\mathrm{\prime }}=\mathrm{Frac}{R}^{\mathrm{\prime }}$. A point of $A(K^{\mathrm{\prime }\mathrm{s}\mathrm{h}})$ uses only finitely many coordinates and relations, so it is defined over the fraction field of a pointed local etale neighbourhood of $R^{\mathrm{\prime }}$. Spread that neighbourhood to an etale scheme $Z^{\mathrm{\prime }}\to Z$ near its special generic point ${\eta }^{\mathrm{\prime }}$, and spread the point to a generic rational map $Z_K^{\mathrm{\prime }}⇢A$. Shrink to the open consisting of the generic fibre and the special component containing ${\eta }^{\mathrm{\prime }}$, so (b) supplies an $R$-rational map into some $V_i$ defined at ${\eta }^{\mathrm{\prime }}$. Localizing and then passing to $R^{\mathrm{\prime }\mathrm{s}\mathrm{h}}$ extends the given point to an $R^{\mathrm{\prime }\mathrm{s}\mathrm{h}}$-section of $(V_i{)}_{R^{\mathrm{\prime }}}$. Thus the base-changed finite collection is weak. The extension $K^{\mathrm{\prime }}\mathrm{/}K$ may have transcendence degree; no finite-separable identification of the two fraction fields is used.