Separated minimal union and translations

Choose finitely many representatives $X_1,\dots ,X_m$ of all minimal equivalence classes using [F1]. For $i\mathrm{\ne }j$, let ${\mathrm{\Gamma }}_{ij}$ be the schematic closure of the generic diagonal $A\hookrightarrow X_i{\times }_R{X}_j$. It is integral and flat over $R$ by [F3]. Suppose its special-fibre support has dense image in $(X_i{)}_k$. The ambient product is regular of dimension $2g+1$, and the generic diagonal has codimension $g$ there; its closure therefore has dimension $g+1$. A component of $({\mathrm{\Gamma }}_{ij}{)}_k$ has dimension at most $g$, since it is a height-one component cut out by the nonzero divisor $\pi$. A component dominating the $g$-dimensional $(X_i{)}_k$ is consequently generically finite over it. At its generic point $q$, the projection $p_i:{\mathrm{\Gamma }}_{ij}\to X_i$ is quasi-finite. It is separated and birational, because its generic-fibre map is the identity. On an affine neighbourhood of $\xi =p_i(q)$, [F5] factors it as an open immersion into a finite scheme. The reduced closure of the birational generic component in that finite scheme is finite birational over the normal coordinate ring of $X_i$ and lies in its fraction field, so normality makes it equal to that coordinate ring. Thus $p_i$ is an isomorphism near $q$ and $\xi$. The other projection then gives an $R$-birational map between $X_i$ and $X_j$; their minimality and [F1] imply they are equivalent, contrary to their representing distinct classes. The same argument applies to the projection to $X_j$. Therefore both special-fibre projection images are nowhere dense. Remove their closures from the special fibres, for all finitely many pairs, and write $X_i^{\circ }$ for the resulting open models. The generic diagonal is now closed in each $X_i^{\circ }{\times }_R{X}_j^{\circ }$ for $i\mathrm{\ne }j$.

Glue the $X_i^{\circ }$ along their common open generic fibre $A$ by the identity. The identity and cocycle conditions hold because every overlap is the same $A$. The resulting scheme $X$ is smooth and of finite type over $R$, since these properties hold on its open cover by the $X_i^{\circ }$. Its diagonal is a closed immersion: on $X_i^{\circ }{\times }_R{X}_i^{\circ }$ this follows from separatedness of $X_i^{\circ }$, and on $X_i^{\circ }{\times }_R{X}_j^{\circ }$ for $i\mathrm{\ne }j$ its image is the closed generic diagonal established in step 1.1; closedness is local on the target by [F4]. Every special fibre remains nonempty after removing nowhere-dense closed subsets, so $X\to \mathrm{Spec}R$ is surjective; it is flat because it is smooth. Thus $X$ is a smooth separated finite-type faithfully flat $R$-model of $A$.

First take $R^{\mathrm{\prime }}=R$. Let $C$ be an irreducible component of the special fibre of $X$, and let $U_C=A\cup C$, an open model with irreducible special fibre. By [F1] it is minimal. Split the finite weak model collection in [F2] into open models with irreducible special fibre; this preserves its weak property. For $a\in A(K)$, apply [F2] to $t_a:A\to A$ to obtain an $R$-rational map $f:U_C⇢Y$ into one such member, with generic fibre $t_a$. Let $\omega$ be the invariant top form on $A$. On the domain of $f$, the pullback of the normalized generator ${\pi }^{-\mathrm{ord}(Y)}\omega$ is ${\pi }^{\mathrm{ord}(U_C)-\mathrm{ord}(Y)}$ times the normalized generator on $U_C$, because $t_a^{\ast }\omega =\omega$. Regularity of this pullback gives $\mathrm{ord}(U_C)\ge \mathrm{ord}(Y)$. Minimality of $U_C$ gives the reverse inequality, so the orders are equal. The pullback of the normalized top form is then a unit, so the relative differential determinant is a unit and $f$ is etale on its domain. It is separated and birational, hence quasi-finite; [F5] and normality of $Y$ show it is an open immersion there. In particular $Y$ is minimal and belongs to one of the finitely many classes represented in step 1.1. Composing with the representative's identity birational map gives an open-immersion extension of $t_a$ into $X$ on that domain. Doing this for each $C$ gives compatible rational maps because their generic restrictions are all $t_a$; they glue to an $R$-rational self-map of $X$. On its domain the glued map still pulls back the normalized invariant top form to a unit, so it is etale; it is birational, and [F5] makes it an open immersion. The same construction for $t_{-a}$ gives the inverse birational map, since the composites agree with the identity on $A$ and [F3] gives dense-open agreement.

For a generic smooth-DVR extension $R^{\mathrm{\prime }}={\mathcal{O}}_{Z,\eta }$, where $Z$ is smooth and of finite type over $R$ and $\eta$ is a generic point of its special fibre, base change the construction to $R^{\mathrm{\prime }}$. The weak model property and the representatives' minimality persist by [F1, F2], after splitting the special fibres into their irreducible components. The argument of step 3.1 therefore applies to each component of $X_{R^{\mathrm{\prime }}}$ and every $a\in A(K^{\mathrm{\prime }})$, where $K^{\mathrm{\prime }}=\mathrm{Frac}(R^{\mathrm{\prime }})$, giving the claimed $R^{\mathrm{\prime }}$-birational open immersion.