In this section we recall some basic facts about the algebraic geometry of toric schemes over a DVR. These toric schemes were introduced in [KKMS73, Chapter IV, §3], and we refer to this reference for more details. They are described and classified in terms of fans in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$. In this section we will mostly consider proper toric schemes over a DVR. As a consequence of Corollary 3.15, proper toric schemes over a DVR can be described and classified in terms of complete SCR polyhedral complexes in $N_{\mathbb{R}}$ as, for instance, in [NS06].

Let $K$ be a field equipped with a nontrivial discrete valuation ${\operatorname{val}}\colon K^{\times}\twoheadrightarrow\mathbb{Z}$. In this section we do not assume $K$ to be complete. As usual, we denote by $K^{\circ}$ the valuation ring, by $K^{\circ\circ}$ its maximal ideal, by $\varpi$ a generator of $K^{\circ\circ}$ and by $k$ the residue field. We assume that ${\operatorname{val}}(\varpi)=1$. We denote by $S$ the base scheme $S=\operatorname{Spec}(K^{\circ})$, by $\eta$ and $o$ the generic and the special points of $S$ and, for a scheme $\mathcal{X}$ over $S$, we set $\mathcal{X}_{\eta}=\mathcal{X}\times_{S}\operatorname{Spec}(K)$ and $\mathcal{X}_{o}=\mathcal{X}\times_{S}\operatorname{Spec}(k)$ for its generic and special fibre respectively. We will denote by $\mathbb{T}_{S}=\mathbb{T}_{K^{0}}\simeq\mathbb{G}_{m,S}^{n}$ a split torus over $S$. Let $\mathbb{T}=\mathbb{T}_{K}$, $N$ and $M$ be as in §4.1. We will write ${\widetilde{N}}=N\oplus\mathbb{Z}$ and ${\widetilde{M}}=M\oplus\mathbb{Z}$.

If ${\mathcal{X}}$ is a toric scheme over $S$, then ${\mathcal{X}}_{\eta}$ is a toric variety over $K$ with torus $\mathbb{T}$.

Since, by definition, a toric scheme is integral and contains $\mathbb{T}$ as a dense open subset, it is flat over $S$. Thus a toric model is a particular case of a model as in Definition 2.11.

Let ${\widetilde{\Sigma}}$ be a fan in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$. To the fan $\widetilde{\Sigma}$ we associate a toric scheme ${\mathcal{X}}_{\widetilde{\Sigma}}$ over $S$. Let $\sigma\in{\widetilde{\Sigma}}$ be a cone and $\sigma^{\vee}\subset{\widetilde{M}}_{\mathbb{R}}$ its dual cone. Set ${\widetilde{M}}_{\sigma}={\widetilde{M}}\cap\sigma^{\vee}$. Let $K^{\circ}[{\widetilde{M}}_{\sigma}]$ be the semigroup $K^{\circ}$-algebra of ${\widetilde{M}}_{\sigma}$. By definition, $(0,1)\in{\widetilde{M}}_{\sigma}$. Thus $(\chi^{(0,1)}-\varpi)$ is an ideal of $K^{\circ}[{\widetilde{M}}_{\sigma}]$. There is a natural isomorphism

(4.57)

$$K^{\circ}[{\widetilde{M}}_{\sigma}]/(\chi^{(0,1)}-\varpi)\simeq\Big\{\sum_{(m,l)\in{\widetilde{M}}_{\sigma}}\alpha_{m,l}\varpi^{l}\chi^{m}\mid\alpha_{m,l}\in K^{\circ}\text{ and, }\overset{\circ}{\forall}(m,l),\alpha_{m,l}=0\Big\}$$

that we use to identify both rings. The ring $K^{\circ}[{\widetilde{M}}_{\sigma}]/(\chi^{(0,1)}-\varpi)$ is an integrally closed domain. We set

$${\mathcal{X}}_{\sigma}=\operatorname{Spec}(K^{\circ}[{\widetilde{M}}_{\sigma}]/(\chi^{(0,1)}-\varpi))$$

for the associated affine toric scheme over $S$. For short we will use the notation

(4.58)

$$K^{\circ}[{\mathcal{X}}_{\sigma}]=K^{\circ}[{\widetilde{M}}_{\sigma}]/(\chi^{(0,1)}-\varpi).$$

For cones $\sigma,\sigma^{\prime}\in{\widetilde{\Sigma}}$, with $\sigma\subset\sigma^{\prime}$ we have a natural open immersion of affine schemes ${\mathcal{X}}_{\sigma}\hookrightarrow{\mathcal{X}}_{\sigma^{\prime}}$. Using these open immersions as gluing data, we define the scheme

$${\mathcal{X}}_{\widetilde{\Sigma}}=\bigcup_{\sigma\in{\widetilde{\Sigma}}}{\mathcal{X}}_{\sigma}.$$

This is a reduced and irreducible normal scheme of finite type over $S$ of relative dimension $n$.

There are two types of cones in ${\widetilde{\Sigma}}$. The ones that are contained in the hyperplane $N_{\mathbb{R}}\times\{0\}$, and the ones that are not. If $\sigma$ is contained in $N_{\mathbb{R}}\times\{0\}$, then $(0,-1)\in{\widetilde{M}}_{\sigma}$, and $\varpi$ is invertible in $K^{\circ}[{\mathcal{X}}_{\sigma}]$. Therefore $K^{\circ}[{\mathcal{X}}_{\sigma}]\simeq K[M_{\sigma}]$; hence ${\mathcal{X}}_{\sigma}$ is contained in the generic fibre and it agrees with the affine toric variety $X_{\sigma}$. If $\sigma$ is not contained in $N_{\mathbb{R}}\times\{0\}$, then ${\mathcal{X}}_{\sigma}$ is not contained in the generic fibre.

To stress the difference between both types of affine schemes we will follow the following notations. Let $\Pi$ be the SCR polyhedral complex in $N_{\mathbb{R}}$ obtained by intersecting ${\widetilde{\Sigma}}$ by the hyperplane $N_{\mathbb{R}}\times\{1\}$ as in Corollary 3.15, and $\Sigma$ the fan in $N_{\mathbb{R}}$ obtained by intersecting ${\widetilde{\Sigma}}$ with $N_{\mathbb{R}}\times\{0\}$. For $\Lambda\in\Pi$, the cone $\operatorname{c}(\Lambda)\in\widetilde{\Sigma}$ is not contained in $N\times\{0\}$. We will write ${\widetilde{M}}_{\Lambda}={\widetilde{M}}_{\operatorname{c}(\Lambda)}$, $K^{\circ}[{\widetilde{M}}_{\Lambda}]=K^{\circ}[{\widetilde{M}}_{\operatorname{c}(\Lambda)}]$, ${\mathcal{X}}_{\Lambda}={\mathcal{X}}_{\operatorname{c}(\Lambda)}$ and $K^{\circ}[{\mathcal{X}}_{\Lambda}]=K^{\circ}[{\mathcal{X}}_{\operatorname{c}(\Lambda)}]$.

Given polyhedrons $\Lambda,\Lambda^{\prime}\in\Pi$, with $\Lambda\subset\Lambda^{\prime}$, we have a natural open immersion of affine toric schemes ${\mathcal{X}}_{\Lambda}\hookrightarrow{\mathcal{X}}_{\Lambda^{\prime}}$. Moreover, if a cone $\sigma\in\Sigma$ is a face of a cone $\operatorname{c}(\Lambda)$ for some $\Lambda\in\Pi$, then the affine toric variety $X_{\sigma}$, is also an open subscheme of ${\mathcal{X}}_{\Lambda}$. The open cover (4.58) can be written as

$${\mathcal{X}}_{\widetilde{\Sigma}}=\bigcup_{\Lambda\in\Pi}{\mathcal{X}}_{\Lambda}\cup\bigcup_{\sigma\in\Sigma}X_{\sigma}.$$

We will reserve the notation ${\mathcal{X}}_{\Lambda}$, $\Lambda\in\Pi$ for the affine toric schemes that are not contained in the generic fibre and denote by $X_{\sigma}$, $\sigma\in\Sigma$ the affine toric schemes contained in the generic fibre, because they are toric varieties over $K$.

The scheme ${\mathcal{X}}_{0}$ corresponding to the polyhedron $0:=\{0\}$ is a group $S$-scheme which is canonically isomorphic to $\mathbb{T}_{S}$. The $S$-action of $\mathbb{T}_{S}$ over ${\mathcal{X}}_{\widetilde{\Sigma}}$ is constructed as in the case of varieties over a field. Moreover there are open immersions $\mathbb{T}_{K}\hookrightarrow{\mathcal{X}}_{\eta}\hookrightarrow{\mathcal{X}}_{\widetilde{\Sigma}}$ of schemes over $S$ and the action of $\mathbb{T}_{S}$ on ${\mathcal{X}}_{\widetilde{\Sigma}}$ extends the action of $\mathbb{T}_{K}$ on itself. Thus ${\mathcal{X}}_{\widetilde{\Sigma}}$ is a toric scheme over $S$. Moreover, the fan $\Sigma$ defines a toric variety over $K$ which coincides with the generic fibre ${\mathcal{X}}_{\widetilde{\Sigma},\eta}$. Thus, ${\mathcal{X}}_{\widetilde{\Sigma}}$ is a toric model of $X_{\Sigma}$. The special fibre ${\mathcal{X}}_{\widetilde{\Sigma},o}={\mathcal{X}}_{\widetilde{\Sigma}}\underset{S}{\times}\operatorname{Spec}(k)$ has an induced action by $\mathbb{T}_{k}$, but, in general, it is not a toric variety over $k$, because it is not irreducible nor reduced. The reduced schemes associated to its irreducible components are toric varieties over $k$ with this action.

Every toric scheme over $S$ can be obtained by the above construction. Indeed, this construction gives a classification of toric schemes by fans in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$ [KKMS73, §IV.3(e)].

If the fan ${\widetilde{\Sigma}}$ is complete, then the scheme ${\mathcal{X}}_{{\widetilde{\Sigma}}}$ is proper over $S$. In this case the set $\{{\mathcal{X}}_{\Lambda}\}_{\Lambda\in\Pi}$ is an open cover of ${\mathcal{X}}_{{\widetilde{\Sigma}}}$. Proper toric schemes over $S$ can also be classified by complete SCR polyhedral complexes in $N_{\mathbb{R}}$. This is not the case for general toric schemes over $S$ as is shown in [BS10].

If we are interested in toric schemes as toric models of a toric variety, we can restate the previous result as follows.

For the rest of the section we will restrict ourselves to the proper case and we will denote by $\Pi$ a complete SCR polyhedral complex. To it we associate a complete fan $\operatorname{c}(\Pi)$ in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$ and a complete fan $\operatorname{rec}(\Pi)$ in $N_{\mathbb{R}}$. For short, we will use the notation

(4.61)

$${\mathcal{X}}_{\Pi}={\mathcal{X}}_{\operatorname{c}(\Pi)},$$

and we will identify the generic fibre ${\mathcal{X}}_{\Pi,\eta}$ with the toric variety $X_{\operatorname{rec}(\Pi)}$.

This example can be generalized to any complete fan $\Sigma$ in $N_{\mathbb{R}}$.

The description of toric orbits in the case of a toric scheme over a DVR is more involved than the case of toric varieties over a field, because we have to consider two kind of orbits.

In the first place, there is a bijection between $\operatorname{rec}(\Pi)$ and the set of orbits under the action of $\mathbb{T}_{K}$ on ${\mathcal{X}}_{\Pi,\eta}$, that sends a cone $\sigma\in\operatorname{rec}(\Pi)$ to the orbit $O(\sigma)\subset{\mathcal{X}}_{\Pi,\eta}=X_{\operatorname{rec}(\Pi)}$ as in the case of toric varieties over a field. We will denote by ${\mathcal{V}}(\sigma)$ the Zariski closure in ${\mathcal{X}}_{\Pi}$ of the orbit $O(\sigma)$ with its structure of reduced closed subscheme. Then ${\mathcal{V}}(\sigma)$ is a horizontal $S$-scheme, in the sense that the structure morphism ${\mathcal{V}}(\sigma)\to S$ is dominant, of relative dimension $n-\dim(\sigma)$.

Next we describe ${\mathcal{V}}(\sigma)$ as a toric scheme over $S$. As before, we write $N(\sigma)=N/(N\cap\mathbb{R}\sigma)$ and let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the linear projection. Each polyhedron $\Lambda$ such that $\sigma\subset\operatorname{rec}(\Lambda)$ defines a polyhedron $\pi_{\sigma}(\Lambda)$ in $N(\sigma)_{\mathbb{R}}$. One verifies that these polyhedra form a complete SCR polyhedral complex in $N(\sigma)_{\mathbb{R}}$, that we denote $\Pi(\sigma)$. This polyhedral complex is called the star of $\sigma$ in $\Pi$.

In the second place, there is a bijection between $\Pi$ and the set of orbits under the action of $\mathbb{T}_{k}$ on ${\mathcal{X}}_{o}$ over the closed point $o$. Given a polyhedron $\Lambda\in\Pi$, we set

$${\widetilde{N}}(\Lambda)={\widetilde{N}}/({\widetilde{N}}\cap\mathbb{R}\negthinspace\operatorname{c}(\Lambda)),\quad{\widetilde{M}}(\Lambda)={\widetilde{N}}(\Lambda)^{\vee}={\widetilde{M}}\cap\operatorname{c}(\Lambda)^{\bot}.$$

We denote $O(\Lambda)=\operatorname{Spec}(k[{\widetilde{M}}(\Lambda)])$.This is a torus over the residue field $k$ of dimension $n-\dim(\Lambda)$. There is a surjection of rings

$$K^{\circ}[{\widetilde{M}}_{\Lambda}]\longrightarrow k[{\widetilde{M}}(\Lambda)],\quad\chi^{(m,l)}\longmapsto\begin{cases}\chi^{(m,l)}&\text{ if }(m,l)\in{\widetilde{M}}(\Lambda),\\ 0&\text{ if }(m,l)\notin{\widetilde{M}}(\Lambda).\end{cases}$$

Since the element $(0,1)$ does not belong to ${\widetilde{M}}(\Lambda)$, then this surjection sends the ideal $(\chi^{(0,1)}-\varpi)$ to zero. Therefore, it factorizes through a surjection $K^{\circ}[{\mathcal{X}}_{\Lambda}]\to k[{\widetilde{M}}(\Lambda)]$, that defines a closed immersion $O(\Lambda)\hookrightarrow{\mathcal{X}}_{\Lambda}$. The subscheme $O(\Lambda)$ is contained in the special fibre ${\mathcal{X}}_{\Pi,o}$, because the surjection sends $\varpi$ to zero. By this reason, the orbits of this type will be called vertical.

We will denote by $V(\Lambda)$ the Zariski closure of the orbit $O(\Lambda)$. Then, $V(\Lambda)$ is a vertical cycle in the sense that its image by the structure morphism is the closed point $o$. We next describe its toric structure. For each polyhedron $\Lambda^{\prime}$ such that $\Lambda$ is a face of $\Lambda^{\prime}$, the image of $\operatorname{c}(\Lambda^{\prime})$ under the projection $\pi_{\Lambda}\colon{\widetilde{N}}_{\mathbb{R}}\to{\widetilde{N}}(\Lambda)_{\mathbb{R}}$ is a strongly convex rational cone that we denote $\sigma_{\Lambda^{\prime}}$. The cones $\sigma_{\Lambda^{\prime}}$ form a fan of ${\widetilde{N}}(\Lambda)_{\mathbb{R}}$ that we denote $\Pi(\Lambda)$. Observe that the fan $\Pi(\Lambda)$ is the analogue of the star of a cone defined in (4.5). For each cone $\sigma\in\Pi(\Lambda)$ there is a unique polyhedron $\Lambda_{\sigma}\in\Pi$ such that $\Lambda$ is a face of $\Lambda_{\sigma}$ and $\sigma=\pi_{\Lambda}(\operatorname{c}(\Lambda_{\sigma}))$.

The description of the adjacency relations between orbits is similar to the one for toric varieties over a field. The orbit $V(\Lambda)$ is contained in $V(\Lambda^{\prime})$ if and only if the polyhedron $\Lambda^{\prime}$ is a face of the polyhedron $\Lambda$. Similarly, ${\mathcal{V}}(\sigma)$ is contained in ${\mathcal{V}}(\sigma^{\prime})$ if and only if $\sigma^{\prime}$ is a face of $\sigma$. Finally, $V(\Lambda)$ is contained in ${\mathcal{V}}(\sigma)$ if and only if $\sigma$ is a face of the cone $\operatorname{rec}(\Lambda)$.

To each polyhedron $\Lambda\in\Pi$, hence to each vertical orbit, we can associate a combinatorial invariant, which we call its multiplicity. For a vertex $v\in\Pi^{0}$, this invariant agrees with the order of vanishing of $\varpi$ along the component $V(v)$ (see (4.87)).

Denote by $\operatorname{\jmath}\colon N\to{\widetilde{N}}$ the inclusion $\operatorname{\jmath}(u)=(u,0)$ and by $\operatorname{pr}\colon{\widetilde{M}}\to M$ the projection $\operatorname{pr}(m,l)=m$. We identify $N$ with its image. We set

$$N(\Lambda)=N/(N\cap\mathbb{R}\negthinspace\operatorname{c}(\Lambda)),\quad M(\Lambda)=M\cap\operatorname{pr}(\operatorname{c}(\Lambda)^{\bot}).$$

Then $\operatorname{\jmath}$ and $\operatorname{pr}$ induce inclusions of lattices of finite index $N(\Lambda)\to{\widetilde{N}}(\Lambda)$ and ${\widetilde{M}}(\Lambda)\to M(\Lambda)$, that we denote also by $\operatorname{\jmath}$ and $\operatorname{pr}$, respectively. These inclusions are dual of each other and in particular, their indexes agree.

We now discuss equivariant morphisms of toric schemes.

It can be verified that a toric morphism of schemes over $S$ is also equivariant. In the sequel, we extend the construction of equivariant morphisms in §4.2 to proper toric schemes. Before that, we need to relate rational points on the open orbit of the toric variety with lattice points in $N$.

Let $\mathbb{T}_{S,i}$, $i=1,2$, be split tori over $S$. For each $i$, let $N_{i}$ be the corresponding lattice and $\Pi_{i}$ a complete SCR polyhedral complex in $N_{i,\mathbb{R}}$. Let $A\colon N_{1}\to N_{2}$ be an affine map such that, for every $\Lambda_{1}\in\Pi_{1}$, there exists $\Lambda_{2}\in\Pi_{2}$ with $A(\Lambda_{1})\subset\Lambda_{2}$. Let $p\in{\mathcal{X}}_{\Pi_{2},0}(K)=\mathbb{T}_{2}(K)$ such that ${\operatorname{val}}(p)=A(0)$. Write $A=H+{\operatorname{val}}(p)$, where $H\colon N_{1}\to N_{2}$ is a linear map. $H$ induces a morphism of algebraic groups

$$\rho_{H}\colon\mathbb{T}_{S,1}\longrightarrow\mathbb{T}_{S,2}.$$

Let $\Sigma_{i}=\operatorname{rec}(\Pi_{i})$. For each cone $\sigma_{1}\in\Sigma_{1}$, there exists a cone $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$. Therefore $H$ and $p$ define an equivariant morphism $\varphi_{p,H}\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ of toric varieties over $K$ as in Theorem 4.9.

As an example of the above construction, we consider the toric subschemes associated to orbits under the action of subtori. Let $N$ be a lattice, $\Pi$ a complete SCR polyhedral complex in $N_{\mathbb{R}}$ and set $\Sigma=\operatorname{rec}(\Pi)$. Let $Q\subset N$ be a saturated sublattice and let $p\in X_{\Sigma,0}(K)$. We set $u_{0}={\operatorname{val}}(p)$. We consider the affine map $A\colon Q_{\mathbb{R}}\to N_{\mathbb{R}}$ given by $A(v)=v+u_{0}$. Recall that the sublattice $Q$ and the point $p$ induce maps of toric varieties (4.11)

$$X_{\Sigma_{Q}}\longrightarrow Y_{\Sigma_{Q},p}\lhook\joinrel\longrightarrow X_{\Sigma}.$$

We want to identify the toric model of $X_{\Sigma_{Q}}$ induced by the toric model ${\mathcal{X}}_{\Pi}$ of $X_{\Sigma}$. We define the complete SCR polyhedral complex $\Pi_{Q,u_{0}}=A^{-1}\Pi$ of $Q_{\mathbb{R}}$. Then, $\operatorname{rec}(\Pi_{Q,u_{0}})=\Sigma_{Q}$. Applying the construction of Proposition 4.72, we obtain an equivariant morphism of schemes over $S$

(4.73)

$${\mathcal{X}}_{\Pi_{Q,u_{0}}}\longrightarrow{\mathcal{X}}_{\Pi}.$$

The image of this map is the Zariski closure of $Y_{\Sigma_{Q},p}$ and ${\mathcal{X}}_{\Pi_{Q,u_{0}}}$ is a toric model of $X_{\Sigma_{Q}}$. This map will be denoted either as $\Phi_{p,A}$ or $\Phi_{p,Q}$. Observe that the abstract toric scheme ${\mathcal{X}}_{\Pi_{Q,u_{0}}}$ only depends on $Q$ and on ${\operatorname{val}}(p)$.