By the definition of a dlt-pair, the scheme $\mathscr{X}$ is regular at the generic point of every stratum of $D$, and at all the other points $x\in\mathscr{X}$, the minimal log discrepancy $\mathrm{mld}_{x}(\mathscr{X},D)$ is positive. Since $K_{\mathscr{X}/R}+D$ is Cartier, $\mathrm{mld}_{x}(\mathscr{X},D)$ is an integer, and therefore at least $1$. The inequality

now implies that $\mathscr{X}$ is terminal. In particular, it is regular in codimension two. ∎

One can construct $(S,s)$ and $\mathscr{Y}$ satisfying (1) and (2) by means of a standard argument based on spreading out and Greenberg approximation; see for instance [MN15, 5.1.2] and the proof of [NX16a, 4.2.4] (note that normality of $\mathscr{Y}$ automatically follows from the fact that $\mathscr{Y}\setminus\mathscr{Y}_{s}$ is normal and $\mathscr{Y}_{s}\cong\mathscr{X}_{k}$ is reduced). In this construction we can also spread out a global generator $\omega$ of the relative canonical line bundle $\omega_{\mathscr{X}/R}$, which then induces a global generator for $\omega_{\mathscr{Y}/S}$, yielding the triviality of $K_{\mathscr{Y}/S}$.

If $N$ is at least $2$, then for every point $x$ of $\mathscr{X}_{k}\cong\mathscr{Y}_{s}$, the model $\mathscr{X}$ is regular at $x$ if and only if $\mathscr{Y}$ is regular at $x$ [MN15, 5.1.2(d)]. Thus the pair $(\mathscr{Y},\mathscr{Y}_{s})$ is snc at all the points of $\mathscr{Y}_{s}$ where $(\mathscr{X},\mathscr{X}_{k})$ is snc. Taking $N$ sufficiently large, we can arrange that every prime component $E$ of $\mathscr{Y}_{s}$ is $\mathbb{Q}$-Cartier in $\mathscr{Y}$. More precisely, let $x$ be a point of $\mathscr{X}_{k}$ and let $m_{x}$ be the Cartier index of $E$ in $\mathscr{X}$ at $x$. Let $f$ be a local generator for the ideal sheaf $\mathcal{O}_{\mathscr{X}}(-m_{x}E)$ at $x$. Assume that $N>m_{x}$ and let $g$ be any element of $\mathcal{O}_{\mathscr{Y},x}$ that is congruent to $f$ modulo $t^{N}$. Obviously, $g$ cannot vanish at any other component of $\mathscr{Y}_{s}$, because $t$ vanishes along each of these components and $f$ does not. On the other hand, $f$ divides $t^{m_{x}}$ in $\mathcal{O}_{\mathscr{X},x}$, so that $g$ divides $t^{m_{x}}$ in $\mathcal{O}_{\mathscr{Y},y}$ since $N>m_{x}$. Thus the zero locus of $g$ is supported in $\mathscr{Y}_{s}$, which means that $g=0$ is a local equation for $m_{x}E$ in $\mathscr{Y}$ at $x$. From now on, we assume that $N$ has been chosen large enough to guarantee that $N\geq 2$ and every prime component of $\mathscr{Y}_{s}$ is $\mathbb{Q}$-Cartier.

Let $E$ be a prime component of $\mathscr{X}_{k}$, denote by $\widetilde{E}$ its normalization, and let $\Delta$ be the pullback of the $\mathbb{Q}$-Cartier divisor $\mathscr{X}_{k}-E$ to $\widetilde{E}$. The scheme $\mathscr{X}$ is regular in codimension two by Lemma 4.2. It follows that the different $\mathrm{Diff}_{\widetilde{E}}(\mathscr{X}_{k}-E)$ coincides with $\Delta$. Thus the pair $(\widetilde{E},\Delta)$ is dlt by adjunction [Ko13, 4.8], using the same reasoning as in the proof of [Ko13, 4.16.4] (except that we have not yet established the normality of $E$). Since $N\geq 2$, the scheme $\mathscr{Y}$ is regular in codimension two, as well; since it is of finite type over $k$, we can apply inversion of adjunction [Ko13, 4.9] to deduce that $(\mathscr{Y},\mathscr{Y}_{s})$ is log canonical on a neighbourhood of $E$, and that the log canonical centers of $(\mathscr{Y},\mathscr{Y}_{s})$ contained in $E$ are precisely the images of the log canonical centers of $(\widetilde{E},\Delta)$. At the generic point of such a log canonical center, the pair $(\mathscr{Y},\mathscr{Y}_{s})$ is snc because the same holds for $(\mathscr{X},\mathscr{X}_{k})$. Varying $E$, we obtain that $(\mathscr{Y},\mathscr{Y}_{s})$ is dlt. This implies that every stratum of $\mathscr{X}_{k}\cong\mathscr{Y}_{s}$ is normal [Ko13, 4.16]; thus, in retrospect, we see that $\widetilde{E}=E$. ∎

In the proof of Proposition 4.3, we have constructed a dlt pair $(\mathscr{Y},\mathscr{Y}_{s})$ with $\mathscr{Y}$ of finite type over $k$ such that there exists an isomorphism of $k$-schemes $\mathscr{X}_{k}\to\mathscr{Y}_{s}$ that identifies the log canonical centers of $(\mathscr{X},\mathscr{X}_{k})$ contained in $\mathscr{X}_{k}$ with those of $(\mathscr{Y},\mathscr{Y}_{s})$ contained in $\mathscr{Y}_{s}$. Thus the result follows from the corresponding properties of $(\mathscr{Y},\mathscr{Y}_{s})$ proven in [Ko13, 4.16]. ∎

We will argue by induction on the dimension of $\mathscr{X}$. The case $\dim(\mathscr{X})=1$ follows at once from the fact that all strata of dlt pairs are normal [Ko13, 4.16(2)]. Thus we may assume that $\dim(\mathscr{X})\geq 2$, and that the result holds for pairs of strictly lower dimension.

Let $x$ be a point on $C$. We claim that every prime divisor in $D$ that contains $C$ is Cartier at $x$. Assuming the claim for now, it follows that $C$ is a local complete intersection at $x$, and thus reduced because it is generically reduced (the pair $(\mathscr{X},D)$ is snc at the generic point of $C$). Now it follows from [Ko13, 4.16(2)] that $C$ is normal, and thus regular since it is of dimension one. But $C$ is defined by the local equations at $x$ of the prime components of $D$ that contain $C$; these local equations form a regular sequence, again by [Ko13, 4.16(2)]. We conclude that locally at $x$, the scheme $\mathscr{X}$ is regular and $D$ is a strict normal crossings divisor.

Thus it suffices to prove our claim. We may assume that $x$ is not a zero-dimensional stratum of $D$, since at such points, the pair $(\mathscr{X},D)$ is snc by the definition of a dlt pair. Let $E$ be a prime divisor in $D$ that contains $C$. Let $F_{1},\ldots,F_{r}$ be the non-empty intersections of $E$ with the other components of $D$, and set $\Delta=F_{1}+\ldots+F_{r}$. Then the pair $(E,\Delta)$ is dlt, and

is Cartier (see Proposition 4.5 and Claim 4.16.4 in [Ko13]). By the induction hypothesis, $E$ is regular at $x$.

Let $m\geq 1$ be the index of $E$ at $x$, that is, the smallest positive integer such that $mE$ is Cartier at $x$. Working locally around $x$, we may assume that $E$ is regular and that $\mathcal{O}_{\mathscr{X}}(mE)$ is a trivial line bundle. The choice of a trivialisation determines a ramified $\mu_{m}$-cover $h\colon\widetilde{\mathscr{X}}\to\mathscr{X}$ defined by

Here $\mathcal{O}_{\mathscr{X}}(-aE)$ is the rank one reflexive sheaf associated with the Weil divisor $-aE$. This is the so-called index one cover of the pair $(\mathscr{X},E)$ at the point $x$; see [KM98, 2.52] for details. The morphism $h$ is étale over all the points where $E$ is Cartier; in particular, it is étale over all the codimension one points of $E$, since $\mathscr{X}$ is regular in codimension two by Lemma 4.2. The minimality of $m$ implies that the inverse image of $x$ in $\widetilde{\mathscr{X}}$ consists of a unique point, which we denote by $\widetilde{x}$.

We write $\widetilde{E}$ for the inverse image of $E$ on $\widetilde{\mathscr{X}}$, and $\widetilde{D}$ for the inverse image of the divisor $D$. By [KM98, 5.20], the pair $(\widetilde{\mathscr{X}},\widetilde{D})$ is log canonical, and $\mathrm{mld}_{\widetilde{x}}(\widetilde{\mathscr{X}},\widetilde{D})$ is positive. Since we chose $x$ on a one-dimensional stratum $C$, the divisor $D$ has $\dim(\mathscr{X})-1$ irreducible components that pass through $x$. This implies that $\widetilde{E}$ is unibranch at $\widetilde{x}$. Otherwise, étale-locally around $\widetilde{x}$, the divisor $\widetilde{D}$ would have at least $\dim(\mathscr{X})$ components passing through $\widetilde{x}$, and $\widetilde{x}$ would be their intersection; but this implies that $\widetilde{x}$ is a log canonical center of $(\widetilde{X},\widetilde{D})$, by [Ko13, 4.41(2)], contradicting the positivity of $\mathrm{mld}_{\widetilde{x}}(\widetilde{\mathscr{X}},\widetilde{D})$.

We denote by $\widetilde{E}^{\prime}$ the normalization of $\widetilde{E}$. Since $\widetilde{E}$ is unibranch at $\widetilde{x}$, there is a unique point $\widetilde{x}^{\prime}$ on $\widetilde{E}^{\prime}$ that lies above $\widetilde{x}\in\widetilde{E}$. We have already observed that the morphism $\widetilde{E}\to E$ induced by $h$ is étale in codimension one; then the normality of $E$ implies that $\widetilde{E}$ is normal in codimension one. Thus $\widetilde{E}^{\prime}\to E$ is also étale in codimension one. Since $E$ is regular, the purity of the branch locus now implies that the finite morphism $\widetilde{E}^{\prime}\to E$ is étale at $\widetilde{x}^{\prime}$; but $\widetilde{x}^{\prime}$ is the unique point that lies above $x\in E$, so that $\widetilde{E}^{\prime}\to E$, and hence $\widetilde{E}\to E$, are isomorphisms. We finally conclude that $m=1$, so that $E$ is Cartier at $x$. ∎

The property that the pair $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ is snc at a point of $\mathscr{X}_{k}$ only depends on the reduction of $\mathscr{X}$ modulo $t^{2}$. Thus by means of the approximation result in Proposition 4.3, we can reduce to the case where the model $\mathscr{X}$ is defined over a smooth algebraic $k$-curve; then the result follows from Theorem 4.5. ∎