We will construct a regular toric $R$-scheme $\mathscr{Y}$ such that $\mathscr{Y}_{k}$ is a strict normal crossings divisor that has a stratum $D$ satisfying $\widehat{\mathscr{X}_{/C}}\cong\widehat{\mathscr{Y}_{/D}}$. Let $\iota$ be the greatest common divisor of the multiplicities $N_{i}$ with $i\in J\cup\{0\}$. We choose lattice vectors $u_{i},\,i\in J\cup\{0\}$ in $\mathbb{Z}^{J}$ with the following property: if we set $v_{0}=(u_{0},N_{0}/\iota)$ and $v_{j}=(u_{j},N_{j}/\iota)$ in $\mathbb{Z}^{J}\oplus\mathbb{Z}$, for all $j\in J$, then the set $\{v_{i},\,i\in J\cup\{0\}\,\}$ is a basis for $\mathbb{Z}^{J}\oplus\mathbb{Z}$. Now we set

in $\mathbb{Z}^{J}\oplus\mathbb{Z}$. Because of the relation (5.3), the last coordinate of $v_{\infty}$ equals $N_{\infty}/\iota$.

For every $i$ in $J\cup\{0,\infty\}$, let $\rho_{i}$ be the ray in $\mathbb{R}^{J}\times\mathbb{R}_{\geq 0}$ spanned by the primitive vector $v_{i}$. Consider the cones $\sigma_{0}$ and $\sigma_{\infty}$ spanned by the rays $\rho_{j}$, $j\in J$ and by $\rho_{0}$ and $\rho_{\infty}$, respectively. The intersection of these cones is the common face spanned by the rays $\rho_{j}$, $j\in J$. Let $\Sigma$ be the fan in $\mathbb{R}^{J}\times\mathbb{R}_{\geq 0}$ with maximal cones $\sigma_{0}$ and $\sigma_{\infty}$. Then $\Sigma$ defines a toric $k$-variety $Y$. We consider the toric morphism

associated with the morphism of cocharacter modules

and we set $\mathscr{Y}=Y\times_{k[t]}R$.

The scheme $\mathscr{Y}$ is regular because the cones $\sigma_{0}$ and $\sigma_{\infty}$ are simple. Moreover, $\mathscr{Y}_{k}$ is a strict normal crossings divisor whose prime components correspond to the rays of $\Sigma$, with multiplicities given by $\iota$ times the last coordinates of the primitive generators of the rays; thus we can write

Set $D=\cap_{j\in J}F_{j}$ and write $d_{0},\,d_{\infty}$ for the intersection points of $D$ with $F_{0}$ and $F_{\infty}$, respectively. By [Fu93, §5.1], we have $D\cdot F_{j}=-b_{j}$ for every $j\in J$.

We will now construct an isomorphism of formal $R$-schemes

For every $n\geq 0$, we denote by $(\mathscr{X}/C)_{n}$ the degree $n$ thickening of $C$ in $\mathscr{X}$, that is, the closed subscheme of $\mathscr{X}$ defined by the $(n+1)$-th power of the defining ideal of $C$. Thus $(\mathscr{X}/C)_{0}=C$ and, by definition, $\widehat{\mathscr{X}_{/C}}$ is the direct limit of the schemes $(\mathscr{X}/C)_{n}$ in the category of locally topologically ringed spaces. For every $j\in J$, denote by $\mathcal{L}_{j}$ the line bundle on $\widehat{\mathscr{X}_{/C}}$ induced by $\mathcal{O}_{\mathscr{X}}(-E_{j}-b_{j}E_{\infty})$. Since the restriction of $\mathcal{L}_{j}$ to $C\cong\mathbb{P}^{1}_{k}$ has degree $0$, we can choose a non-zero global section $s_{j}$ of $\mathcal{L}_{j}|_{C}$. The conormal bundle of $C$ in $\mathscr{X}$ is given by

which is a direct sum of ample line bundles, by our assumption that the numbers $b_{j}$ are all positive. This implies that the degree one cohomology of the conormal line bundle vanishes, so that the maps

are surjective for all $n\geq 0$. Thus we can lift $s_{j}$ to a global section of $\mathcal{L}_{j}$ on $\widehat{\mathscr{X}_{/C}}$, which we will still denote by $s_{j}$. The same argument produces a nowhere vanishing section $s_{0}$ of $\mathcal{O}_{\mathscr{X}}(E_{\infty}-E_{0})$ on $\widehat{\mathscr{X}_{/C}}$; its inverse $s_{\infty}=1/s_{0}$ is a nowhere vanishing global section of $\mathcal{O}_{\mathscr{X}}(E_{0}-E_{\infty})$ on $\widehat{\mathscr{X}_{/C}}$.

Consider the open formal subschemes

of $\widehat{\mathscr{X}_{/C}}$ and $\widehat{\mathscr{Y}_{/D}}$. Note that $s_{i}$ is a global equation for $E_{i}$ on $\mathfrak{X}_{0}$, for every $i\in J\cup\{0\}$. Likewise, $s_{\infty}$ defines $E_{\infty}$ on $\mathfrak{X}_{\infty}$, and $s_{j}s^{b_{j}}_{\infty}$ defines $E_{j}$ on $\mathfrak{X}_{\infty}$, for every $j\in J$. Moreover, $w^{\prime}=ts_{0}^{-N_{0}}\prod_{j\in J}s_{j}^{-N_{j}}$ is an invertible regular function on $\widehat{\mathscr{X}_{/C}}$. Since $C$ is proper, $w^{\prime}$ is constant on $C$, and, in particular, it has a $\iota$-th root; Hensel’s lemma then implies that we can find a regular function $w$ on $\widehat{\mathscr{X}_{/C}}$ such that $w^{\prime}=w^{\iota}$.

Let $\{v_{0}^{\vee},v_{j}^{\vee}\,(j\in J)\}$ be the dual basis of $\{v_{0},v_{j}\,(j\in J)\}$. Then we have

Choose integers $\alpha_{0}$ and $\alpha_{j},\,j\in J$ such that $\alpha_{0}N_{0}+\sum_{j\in J}\alpha_{j}N_{j}=1$. Let $f_{0}\colon\mathfrak{X}_{0}\to\mathfrak{Y}_{0}$ be the morphism of formal $R$-schemes defined by the morphism of topological $R$-algebras

Let $\mathscr{J}$ be the largest ideal of definition on $\mathfrak{Y}_{0}$. Then $\mathscr{J}(\mathfrak{Y}_{0})$ is generated by $\chi^{v_{j}^{\vee}},\,j\in J$. The ideal $\mathscr{J}\mathcal{O}_{\mathfrak{X}_{0}}$ is the largest ideal of definition on $\mathfrak{X}_{0}$, and its global sections are generated by $s_{j},\,j\in J$. In particular, $f_{0}$ is adic. The morphism

is an isomorphism. It follows from [EGA3.1, 4.8.10] that $f_{0}$ is a closed immersion; since $\mathfrak{X}_{0}$ and $\mathfrak{Y}_{0}$ has the same dimension and $\mathfrak{Y}_{0}$ is integral, $f_{0}$ is an isomorphism.

Finally, we consider the second pair of affine charts $\mathfrak{X}_{\infty},\,\mathfrak{Y}_{\infty}$. The lattice vectors $\{-v_{0}^{\vee},v_{j}^{\vee}+b_{j}v^{\vee}_{0}\,(j\in J)\}$ form the dual basis of $\{v_{\infty},v_{j}\,(j\in J)\}$, and

Let $f_{\infty}\colon\mathfrak{X}_{\infty}\to\mathfrak{Y}_{\infty}$ be the morphism of formal $R$-schemes defined by the morphism of topological $R$-algebras $\mathcal{O}(\mathfrak{Y}_{0})\to\mathcal{O}(\mathfrak{X}_{0})$ that maps $\chi^{-v^{\vee}_{0}}$ to $s_{\infty}$ and $\chi^{v_{j}^{\vee}+b_{j}v^{\vee}_{0}}$ to $s_{j}s^{-b_{j}}_{\infty}$, for all $j$ in $J$. By the same reasoning as above, one sees that $f_{\infty}$ is an isomorphism. By construction, it agrees with $f_{0}$ on the intersection of $\mathfrak{X}_{0}$ and $\mathfrak{X}_{\infty}$, and the isomorphisms $f_{0}$ and $f_{\infty}$ glue to an isomorphism of formal $R$-schemes

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