We set $J=\{0,1,\ldots,n-r\}$ such that $Z=\cap_{j\in J}D_{j}$. Since for every irreducible component $D$ of $\mathscr{X}_{k}$, the intersection $D\cap Z$ is connected by assumption, this allows us to denote by $D_{l}$ with $l\in L$ the components of $\mathscr{X}_{k}$ intersecting $Z$ transversally along $Z_{l}\coloneqq Z\cap D_{l}$, so that the toric boundary of $Z$ is given by $\Delta_{Z}=\sum_{l\in L}Z_{l}$.

We denote by $\Sigma\subset N_{\mathbb{R}}$ the fan of $Z$. Its rays are given by $\mathbb{R}_{\geqslant 0}u_{l}$ for $l\in L$, with primitive generators $u_{l}$; the maximal cones of $\Sigma$ are in bijection with the set of unordered $r$-tuples $\{i_{1},\ldots,i_{r}\}\in L^{r}$ such that $\cap_{\beta=1}^{r}D_{i_{\beta}}\cap Z\neq\varnothing$. For a maximal cone $\sigma$ of $\Sigma$, we write $L_{\sigma}\coloneqq\{l\in L\,|\,u_{l}\in\sigma\}$.

Let $\mathcal{N}\coloneqq\nu_{Z/\mathscr{X}}\xrightarrow{p}Z$ be the normal bundle of $Z$ in $\mathscr{X}$, and denote by $Z\subset\mathcal{N}$ the zero section. We write $\mathcal{O}_{\mathscr{X}}(D_{j})_{|Z}=\mathcal{O}_{Z}(F_{j})$ so that $\mathcal{N}=\oplus_{j\in J}\mathcal{O}_{Z}(F_{j})$. Since any Cartier divisor on $Z$ is linearly equivalent to a toric one, for any $j=1,\ldots,n-r$, there exist integers $\lambda_{j,l}$ such that

(2.1.4)

$$\mathcal{O}_{Z}(F_{j})=\mathcal{O}_{Z}\big(-\sum_{l\in L}\lambda_{j,l}Z_{l}\big).$$

For $j=0$ we set $\lambda_{0,l}\coloneqq 1-\sum_{j\in J\setminus\{0\}}\lambda_{j,l}$ and verify that

$\displaystyle\mathcal{O}_{Z}(F_{0})=\mathcal{O}_{\mathscr{X}}(D_{0}-\mathscr{X}_{k})_{|Z}=\mathcal{O}_{Z}\Big(-\sum_{j\in J\setminus\{0\}}F_{j}-\sum_{l\in L}Z_{l}\Big)=\mathcal{O}_{Z}(-\sum_{l\in L}\lambda_{0,l}Z_{l}).$

We obtain that $\mathcal{N}=\oplus_{j\in J}\mathcal{O}_{Z}\big(-\sum_{l\in L}\lambda_{j,l}Z_{l}\big)$ and for all $l$ in $L$

(2.1.5)

$$\sum_{j\in J}\lambda_{j,l}=1.$$

The normal bundle $\mathcal{N}$ is a toric variety of dimension $n+1$. The corresponding fan $\hat{\Sigma}$ lies in $N_{\mathbb{R}}\times\mathbb{R}^{J}$ and consists of the following cones and their faces (see [CLS11, §7.3] for a reference). Let $e_{0},\ldots,e_{n-r}$ be the standard basis of $\mathbb{R}^{J}$; given a cone $\sigma\in\Sigma$, we have

$$\hat{\sigma}=\textrm{Cone}((0,e_{0}),\ldots,(0,e_{n-r}),(u_{l},(\lambda_{j,l}))\,|\,u_{l}\in\sigma)\in\hat{\Sigma}.$$

In particular, we denote the rays of $\hat{\Sigma}$ by

$$v_{j}=(0,e_{j})\textrm{ for }j\in J,\quad v_{l}=(u_{l},(\lambda_{j,l}))\textrm{ for }l\in L.$$

The map

$$\ord(t):N_{\mathbb{R}}\times\mathbb{R}^{J}\rightarrow\mathbb{R}_{\geqslant 0}\quad(u,w)\mapsto\sum_{j=0}^{n-r}w_{j}$$

is $\mathbb{Z}$-linear, sends all the primitive generators of the rays of $\hat{\Sigma}$ to $1$ by Eq. 2.1.5, and is compatible with $\hat{\Sigma}$ and the fan of $\mathbb{A}^{1}_{k}$. Thus, it induces a toric morphism $t:\mathcal{N}\rightarrow\mathbb{A}^{1}_{k}$ whose fiber over $0$ is the toric boundary of $\mathcal{N}$. The base change $\mathscr{N}\coloneqq\mathcal{N}\times_{\mathbb{A}^{1}}R$ to $R$ is a toric $R$-scheme, whose generic fiber is isomorphic to $\mathbb{G}_{m,K}^{n}$. The special fiber $\mathscr{N}_{k}$ can be written as $\mathscr{N}_{k}=\sum_{i\in J\cup L}E_{i}$, where the combinatoric of intersections between components is exactly the same as in $\mathscr{X}_{k}$.

We prove Theorem B by constructing a formal isomorphism

$$f:\widehat{\mathscr{X}_{/Z}}\xrightarrow{\simeq}\widehat{\mathscr{N}_{/Z}}.$$

More specifically, we proceed as follows. We set the notations $\mathfrak{X}=\widehat{\mathscr{X}_{/Z}}$ and $\mathfrak{N}=\widehat{\mathscr{N}_{/Z}}$.

• •

  (Sections 2.2 and 2.3) Let $\sigma\in\Sigma$ be a maximal cone. Denote by $Z_{\sigma}$ and $\mathcal{N}_{\sigma}\coloneqq\mathcal{N}_{Z_{\sigma}/\mathscr{X}}$ the corresponding toric affine charts in $Z$ and $\mathcal{N}$ respectively. This induces an open formal subscheme of $\mathfrak{N}$, which we denote by $\mathfrak{N}_{\sigma}$. We construct a morphism

  $$f_{\sigma}:\mathfrak{X}\setminus\big(\cup_{l\in L\setminus L_{\sigma}}D_{l}\big)\eqqcolon\mathfrak{X}_{\sigma}\,\rightarrow\mathfrak{N}_{\sigma},$$

  in a similar manner to [NXY19]: we construct $n+1$ divisors $W^{\sigma}_{j}$ and $W^{\sigma}_{i}$ on $\mathscr{X}$, whose defining equations on the chart $\mathfrak{X}_{\sigma}$ yields the morphism $f_{\sigma}$. The equations are induced by sections of $\mathcal{O}_{Z}(W^{\sigma}_{j})$ and $\mathcal{O}_{Z}(W^{\sigma}_{i})$: these are first constructed on $Z$, then extended to $\mathfrak{X}$ by the nef condition on the conormal bundle $\nu_{Z/\mathscr{X}}^{*}$ assumed in Theorem B, which ensures the vanishing of higher cohomology groups for the tensor powers of $\nu_{Z/\mathscr{X}}^{*}$.

• •

  (Sections 2.4 and 2.5) Let $\sigma$ and $\sigma^{\prime}$ be two maximal cones of $\Sigma$ intersecting along a face of codimension one. We establish relations among the respective divisors and construct sections on $\mathfrak{X}_{\sigma^{\prime}}$ from those on $\mathfrak{X}_{\sigma}$. This allows us to prove that the morphisms $f_{\sigma}$ on the charts $\mathfrak{X}_{\sigma}$’s can be chosen so that they are compatible on the overlaps $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$. This yields a well defined morphism $f$ which extends the identity on $Z$ and preserves the ideal $\mathscr{I}_{Z}$, so that it turns out to be an isomorphism.