Full minimal model embedding

The generic fibre of $X$ is $A=X_K$, and $G_K$ is the group completion of the strict law on $U_K=A$; hence the identity of $A$ defines a rational map $\phi :X⇢G$ over $R$ which on $U$ is the given open immersion $U\hookrightarrow G$. Every height-one point of $X$ lies either in the generic fibre (where $\phi$ is defined, since it is the identity of $A$) or is a generic point of an irreducible component of the special fibre. Since $U$ is $R$-dense, its complement contains no irreducible component of any fibre, so $U$ contains the generic point of every component of the special fibre; therefore $\phi$ is defined at every height-one point of $X$.

The base $R$ is a regular Noetherian scheme, $X$ is smooth over $R$ and $G$ is a smooth separated finite-type $R$-group scheme, so the codimension-one extension criterion [F2] applies to $\phi$ and produces a unique $R$-morphism $\psi :X\to G$ extending $\phi$. On the generic fibre ${\psi }_K$ is the identity of $A$, so $\psi$ is birational.

We show that $\psi$ is etale. Its top differential is a section of the invertible sheaf $\mathrm{Hom}({\psi }^{\ast }{\bigwedge }^g{\mathrm{\Omega }}_{G\mathrm{/}R},{\bigwedge }^g{\mathrm{\Omega }}_{X\mathrm{/}R})$. It is a unit on $U$, where $\psi$ is the given open immersion, and on the generic fibre, where it is the identity. Thus it is nonzero generically, and its zero divisor can only be supported in $X\setminus U$. Step 1.1 shows that $U$ contains every height-one point. Since $X$ is regular and integral, the zero locus of a nonzero section of an invertible sheaf is an effective Cartier divisor of pure codimension one, unless empty. There is no possible codimension-one support, so the section is nowhere zero and the top differential is an isomorphism. The differential criterion [F4] makes $\psi$ etale. This argument requires no global trivialization of the canonical bundle of the preliminary model.

The morphism $\psi$ is separated, because $X$ and $G$ are separated over $R$, and it is quasi-finite: it is etale, hence locally quasi-finite, and it is a morphism of finite type between quasi-compact schemes, hence quasi-compact; a quasi-compact locally quasi-finite morphism is quasi-finite. Being etale and birational, and having reduced integral source and normal target (smooth over the DVR), it is an open immersion: apply Zariski's main factorization [F5] locally on $G$ to write $\psi =g\circ j$ with $j:X\hookrightarrow Z$ an open immersion and $g:Z\to G$ finite. Replace $Z$ by the reduced closure of its birational generic component, which still contains $j(X)$. Its coordinate algebra is a finite integral subalgebra of the common function field containing the normal target algebra, so it equals that target algebra. Thus $g$ is an isomorphism on this component and $\psi$ is an open immersion.

Consequently $\psi$ identifies $X$ with an open subscheme of $G$ containing $U$; since the completion already contains $U$ as an $R$-dense open by [F1], the larger image of $X$ is $R$-dense in $G$. This is the assertion that the descended completion contains the full separated minimal model $X$, not only the strictification $U$, as an $R$-dense open.