Projective modification of a proper integral DVR-scheme, unchanged in codimension one when regular

Choose a finite nonempty affine cover $X={\bigcup }_i{U}_i$, and put $U={\bigcap }_i{U}_i$. Since $X$ is integral, every $U_i$ contains its generic point and $U$ is nonempty dense. Each $U_i$ is finite type over $R$, so finite algebra generators give a closed immersion into an affine space and hence an immersion $U_i\to {\mathbb{P}}_R^{n_i}$. Let $Z_i$ be its integral closure as a subscheme of that projective space, where “closure” means the reduced scheme-theoretic closure of its image, not normalization. Concretely its ideal on any affine chart is the kernel of evaluation in the function field of $U_i$; these prime ideals localize compatibly and define an integral closed subscheme. It contains $U_i$ as a dense open. Form the integral closure $W$ of the diagonal map $U\to {\prod }_i{\mathbb{P}}_R^{n_i}$ in the same sense. The map of $U$ is an immersion, since its multi-diagonal into ${\prod }_i{U}_i$ is closed by separatedness of $X\mathrm{/}R$; therefore $U$ is a dense open of $W$. Each projection factors through $Z_i$. Write $p_i:W\to Z_i$, $V_i=p_i^{-1}(U_i)$, and $Z={\bigcup }_i{V}_i$.

The maps $V_i\to U_i\to X$ agree on intersections, because they agree on the dense open $U$, their source is integral, and the target is separated. They glue to $\pi :Z\to X$. Each $V_i\to U_i$ is proper by [F1]. The inclusion $V_i\subseteq {\pi }^{-1}(U_i)$ is proper over $U_i$ by the proper-source/separated-target assertion, hence closed; it is also dense since it contains $U$, so it equals ${\pi }^{-1}(U_i)$. Thus $\pi$ is proper by target locality. Its image contains the dense $U_i$-subset $U$ and is closed in each $U_i$, so it is surjective. The same argument applied to $U\subseteq {\pi }^{-1}(U)$, whose map to $U$ is the identity, gives ${\pi }^{-1}(U)=U$. The product of projective spaces has its Segre closed embedding in one projective space: in each chart the product coordinates recover the original affine coordinates, and globally the image is cut out by the two-by-two minors of the rank-one coordinate tensor; iteration gives the finite-product embedding. Hence $Z$, open in closed $W$, has an immersion in projective space. Since $Z$ is proper over $R$ and the ambient projective space is separated, that immersion is proper by [F1] and hence closed. Thus $Z$ is projective and integral, and $\pi$ is birational.

If $X$ is regular and $x$ has codimension one, put $A={\mathcal{O}}_{X,x}$, a DVR by [F2], with fraction field $K(X)$. The base change $Z_A$ is integral and proper birational over $A$. Any affine chart meeting its closed fibre has a finite-type coordinate algebra $B\subseteq K(X)$ containing $A$. An element of negative valuation in $B$ would make the uniformizer invertible in $B$, so cannot occur on such a chart; consequently $B=A$. A chart meeting the closed fibre exists by valuative existence in [F1], and any two such charts give sections agreeing at the generic point and hence equal by separatedness. It follows that $Z_A\to \mathrm{Spec}A$ is an isomorphism. This isomorphism spreads to a neighbourhood of $x$: use finitely many affine charts for the finite-type morphism, express the inverse ring maps by finitely many generators, relations and denominators outside the prime of $x$, and shrink to make their compositions equal to the identity. Charts absent after localization are removed by clearing their relation $1=0$; the remaining finitely many charts glue the inverse. Thus the open isomorphism locus contains every codimension-one point.

If $X$ dominates the trait, the same is true of integral $Z$. Its affine coordinate rings are torsion-free over $R$, hence flat by [F2]. Flatness is affine local, so $Z\mathrm{/}R$ is flat. This proves all clauses, retaining AC through [F1]–[F2].