Defect decrease and finite smoothening

Let $Y$ have schematically dense $k^s$-points. On an affine chart with coordinate ring $B$, evaluation at the $k^s$-points embeds $B$ into a product of copies of $k^s$; tensoring with any field extension $l\mathrm{/}k$, every relation involves finitely many coefficients, so the embedding remains injective, and $k^s{\otimes }_{k}l$ is reduced because $k^s$ is separable algebraic over $k$. Hence $B{\otimes }_{k}l$ is reduced and $Y$ is geometrically reduced. Extending to a perfect closure, the function field of each component is separably generated and the differential rank equals the transcendence degree, so the relative Jacobian criterion produces a smooth neighbourhood of every generic point; smoothness descends through field extensions by [F2]. Therefore the smooth locus of $Y$ is dense and open, and the restriction of ${\mathrm{\Omega }}_{X\mathrm{/}R}$ to it is free on a dense open by [F3].

Shrink around a specialization in $U$, so $Y=U$ is smooth of dimension $r$ and ${\mathrm{\Omega }}_{X\mathrm{/}R}{\mathrm{\mid }}_Y$ is free of rank $r+n$. Choose lifts $y_1,\dots ,y_r,z_1,\dots ,z_n$ whose differentials give its basis, with $z_j$ vanishing on $Y$ and the $d{y}_i$ mapping to a basis of ${\mathrm{\Omega }}_{Y\mathrm{/}k}$. Embed $X$ into affine space with these as initial coordinates. Independent rows of the remaining relation differentials cut out a smooth ambient $Z$ of dimension $r+n$ containing $X$, with these differentials as a basis. Locally $Y\subset Z$ has ideal $J=(\pi ,z_1,\dots ,z_n)$: its displayed equations define a smooth subscheme of dimension $r$ containing $Y$, hence agree with $Y$ locally. Put $X=\mathrm{Spec}(C\mathrm{/}I)$ and $Z=\mathrm{Spec}C$. For $f\in I\subset J$ write $f=\pi g+\sum z_j{g}_j$. Since the map ${\mathrm{\Omega }}_{Z\mathrm{/}R}{\mathrm{\mid }}_Y\to {\mathrm{\Omega }}_{X\mathrm{/}R}{\mathrm{\mid }}_Y$ identifies the chosen bases, $df{\mathrm{\mid }}_Y=0$, so every $g_j\in J$. Thus $f=\pi g+h$ with $h\in J^2$. On every liftable section through $Y$, $f(a)=0$ and $h(a)\in {\pi }^2{R}^{\mathrm{s}\mathrm{h}}$; hence $g(a)\in \pi R^{\mathrm{s}\mathrm{h}}$. Schematic density of those specializations gives $g\in J$. Therefore $I\subseteq J^2$.

In the dilatation of $Z$ write $z_j=\pi z_j^{\mathrm{\prime }}$. Since $I\subset J^2$, each relation $f$ becomes ${\pi }^2{f}^{\mathrm{\prime }}$ with $f^{\mathrm{\prime }}$ in the saturated ideal of the dilatation of $X$. Along a lifted section, the Jacobian rows for $f^{\mathrm{\prime }}$ have $y$-entries ${\pi }^{-2}\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{y}_i$ and $z^{\mathrm{\prime }}$-entries ${\pi }^{-1}\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{z}_j$. Choose a maximal-rank minor realizing the old defect in [F1], of size $q=r+n-d$, where $d$ is the generic relative dimension at the section. The corresponding minor of the divided equations is multiplied by ${\pi }^{-2a-b}$, with $a+b=q$ according to its selected coordinate columns. The new defining ideal may have additional generators, so its minimum minor valuation is at most this value: $\delta (a^{\mathrm{\prime }})\le \delta (a)-q$. If $q=0$, the generic closed immersion $X\subset Z$ agrees near the generic section by smoothness and equal dimension; the ideal vanishes locally at the specialization by flatness and schematic density, so the original section is already smooth. Thus positive defect implies $q\ge 1$, proving (a).

For a set $E$ of nonsmooth $R^{\mathrm{s}\mathrm{h}}$-sections, let $Y_1$ be the reduced closure of their specializations. It satisfies the liftable density condition by construction. Let $U_1$ be its dense smooth open where ${\mathrm{\Omega }}_{X\mathrm{/}R}{\mathrm{\mid }}_{Y_1}$ is locally free, and let $E_1$ be the sections specializing there. Repeat with $E\setminus E_1$, obtaining $Y_2\subset Y_1\setminus U_1$, and continue. The dimensions of the nonempty centres strictly decrease, so this gives a finite partition $E=E_1\bigsqcup \cdots \bigsqcup E_t$ with $Y_i$ the specialization closure of $E_i\bigsqcup \cdots \bigsqcup E_t$. In particular $Y_t$ is permissible for the whole current $E$: every section of $E$ meeting it belongs to $E_t$ and specializes in $U_t$. Blowing up $Y_t$ uniquely lifts sections by properness; those through the centre lie in its $\pi$-chart, since the pulled-back ideal contains $\pi$ and is contained in $(\pi )$. Their positive defects decrease by step 3.1, while all sections outside the centre are unaffected.

Use induction first on the uniform maximal defect $D$ from [F1], and within a fixed $D$ on the partition length $t$. The case $D=0$ is smooth. Blow up $Y_t$ as in step 4.1 and apply the $D-1$ induction to its lifted subset $E_t$. All resulting centres stay over $Y_t$ in the nonsmooth locus, so the other $E_i$ are unaffected. Once $E_t$ is smooth, the remaining sections have defect at most $D$ and their canonical partition has length $t-1$, since the modification is an isomorphism away from $Y_t$; apply the second induction. This terminates with finitely many permissible special-fibre blowups, proper and generically isomorphisms, smoothing every section of $E$. Initially take $E$ to be all nonsmooth sections of $X$. Centres always avoid the smooth locus, so initially smooth sections remain smooth. This proves (b) over every DVR, using no completeness, excellence or perfect-residue hypothesis.