The theory of $\mathbb{T}$-Cartier divisors carries over to the case of toric schemes over a DVR. Let ${\mathcal{X}}$ be a toric scheme over $S$ with torus $\mathbb{T}_{S}$. There are two morphisms from $\mathbb{T}_{S}\times{\mathcal{X}}$ to ${\mathcal{X}}$: the toric action, that we denote by $\mu$, and the second projection, that we denote by $\pi_{2}$. A Cartier divisor $D$ on ${\mathcal{X}}$ is called a $\mathbb{T}$-Cartier divisor if $\mu^{\ast}D=\pi_{2}^{\ast}D.$

$\mathbb{T}$-Cartier divisors over a toric scheme can be described combinatorially. For simplicity, we will discuss only the case of proper schemes. So, let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$, and ${\mathcal{X}}_{\Pi}$ the corresponding toric scheme. Let $\psi$ be an H-lattice function on $\Pi$ (Definitions 3.88 and 3.60). Then $\psi$ defines a $\mathbb{T}$-Cartier divisor in a way similar to the one for toric varieties over a field. We recall that the schemes $\{{\mathcal{X}}_{\Lambda}\}_{\Lambda\in\Pi}$ form an open cover of ${\mathcal{X}}_{\Pi}$. Choose a set of defining vectors $\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}$ of $\psi$. Then we set

$$D_{\psi}=\{({\mathcal{X}}_{\Lambda},\varpi^{-l_{\Lambda}}\chi^{-m_{\Lambda}})\}_{\Lambda\in\Pi},$$

where we are using the identification (4.57). The divisor $D_{\psi}$ only depends on $\psi$ and not on a particular choice of defining vectors.

We consider now toric varieties and $\mathbb{T}$-Cartier divisors over $S$ as models of toric varieties and $\mathbb{T}$-Cartier divisors over $K$.

This example can be generalized as follows.

In this section we will mainly use the language of $\mathbb{T}$-Cartier divisors, but in §6 we will prefer the language of toric line bundles.

The following result follows directly form the definitions.

We are interested in proper toric models and equivalence classes because, by Definition 2.17, a proper toric model of $(X_{\Sigma},D_{\Psi})$ induces an algebraic metric on $L_{\Psi}^{{\text{\rm an}}}$. By Proposition 2.18, equivalent toric models define the same algebraic metric.

We can classify proper models of $\mathbb{T}$-Cartier divisors (and therefore of toric line bundles) in terms of H-lattice functions. We first recall the classification of $\mathbb{T}$-Cartier divisors.

We next derive the classification theorem for models of $\mathbb{T}$-Cartier divisors.

The correspondence between $\mathbb{T}$-Cartier divisors and $\mathbb{T}$-Weil divisors has to take into account that we have two types of orbits. Each vertex $v\in\Pi^{0}$ defines a vertical invariant prime Weil divisor $V(v)$ and every ray $\tau\in\operatorname{rec}(\Pi)^{1}$ defines a horizontal prime Weil divisor ${\mathcal{V}}(\tau)$. If $v\in\Pi^{0}$ is a vertex, by Lemma 4.69, its multiplicity $\operatorname{mult}(v)$ is the smallest positive integer $\nu\geq 1$ such that $\nu v\in N$. If $\tau$ is a ray, we denote by $v_{\tau}$ the smallest lattice point of $\tau\setminus\{0\}$.

We next study the restriction of $\mathbb{T}$-Cartier divisors to orbits and their inverse image by equivariant morphisms. Let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$, and $\psi$ an H-lattice function on $\Pi$. Set $\Sigma=\operatorname{rec}(\Pi)$, and $\Psi=\operatorname{rec}(\psi)$. Choose sets of defining vectors $\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}$ and $\{m_{\sigma}\}_{\sigma\in\Sigma}$ for $\psi$ and $\Psi$, respectively.

Let $\sigma\in\Sigma$. We describe the restriction of $D_{\psi}$ to ${\mathcal{V}}(\sigma)$, the closure of a horizontal orbit. As in the case of toric varieties over a field, we first consider the case when $\Psi|_{\sigma}=0$. Recall that ${\mathcal{V}}(\sigma)$ agrees with the toric scheme associated to the polyhedral complex $\Pi(\sigma)$ and that each element of $\Pi(\sigma)$ is the image by $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ of a polyhedron $\Lambda\in\Pi$ with $\sigma\subset\operatorname{rec}(\Lambda)$. The condition $\Psi|_{\sigma}=0$ implies that we can define

(4.88)

$$\psi(\sigma)\colon N(\sigma)_{\mathbb{R}}\longrightarrow\mathbb{R},\quad u+\mathbb{R}\sigma\longmapsto\psi(u+v)$$

for any $v\in\mathbb{R}\sigma$ such that $u+v\in\bigcup_{\operatorname{rec}(\Lambda)\supset\sigma}\Lambda$. The function $\psi(\sigma)$ can also be described in terms of defining vectors. For each $\Lambda\in\Pi$ with $\sigma\subset\operatorname{rec}(\Lambda)$, we will denote ${\overline{\Lambda}}\in\Pi(\sigma)$ for its image by $\pi_{\sigma}$. For each $\Lambda$ as before, the condition $\Psi|_{\sigma}=0$ implies that $m_{\Lambda}\in M(\sigma)$. Hence we define $(m_{{\overline{\Lambda}}},l_{{\overline{\Lambda}}})=(m_{\Lambda},l_{\Lambda})$ for $\Lambda\in\Pi$ with $\operatorname{rec}(\Lambda)\supset\sigma$.

If $\Psi|_{\sigma}\not=0$, then ${\mathcal{V}}(\sigma)$ and $D_{\psi}$ do not intersect properly and we can only restrict $D_{\psi}$ with ${\mathcal{V}}(\sigma)$ up to rational equivalence. To this end, we consider the divisor $D_{\psi-m_{\sigma}}$, that is rationally equivalent to $D_{\psi}$ and intersects properly with ${\mathcal{V}}(\sigma)$. The restriction of this divisor to ${\mathcal{V}}(\sigma)$ corresponds to the H-lattice function $(\psi-m_{\sigma})(\sigma)$ as defined above.

Let now $\Lambda\in\Pi$ be a polyhedron. We will denote by ${\widetilde{\pi}}_{\Lambda}\colon{\widetilde{N}}\to{\widetilde{N}}(\Lambda)$ and $\pi_{\Lambda}\colon N\to N(\Lambda)$ the projections and by ${\widetilde{\pi}}_{\Lambda}^{\vee}\colon{\widetilde{M}}(\Lambda)\to{\widetilde{M}}$ and $\pi_{\Lambda}^{\vee}\colon M(\Lambda)\to M$ the dual maps. We will use the same notation for the linear maps obtained by tensoring with $\mathbb{R}$.

We first assume that $\psi|_{\Lambda}=0$. If $u\in{\widetilde{N}}(\Lambda)_{\mathbb{R}}$, then there exists a polyhedron $\Lambda^{\prime}$ with $\Lambda$ a face of $\Lambda^{\prime}$ and a point $(v,r)\in\operatorname{c}(\Lambda^{\prime})$ that is sent to $u$ under the projection ${\widetilde{\pi}}_{\Lambda}$. Then we set

(4.90)

$$\psi(\Lambda)\colon{\widetilde{N}}(\Lambda)_{\mathbb{R}}\longrightarrow\mathbb{R},\quad u\longmapsto r\psi(v/r)=m_{\Lambda^{\prime}}(v)+rl_{\Lambda^{\prime}}.$$

The condition $\psi|_{\Lambda}=0$ implies that the above equation does not depend on the choice of $(v,r)$.

We can describe also $\psi(\Lambda)$ in terms of defining vectors. For each cone $\sigma\in\Pi(\Lambda)$ let $\Lambda_{\sigma}\in\Pi$ be the polyhedron that has $\Lambda$ as a face and such that $\operatorname{c}(\Lambda)$ is mapped to $\sigma$ by ${\widetilde{\pi}}_{\Lambda}$. The condition $\psi|_{\Lambda}=0$ implies that $(m_{\Lambda_{\sigma}},l_{\Lambda_{\sigma}})\in{\widetilde{M}}(\Lambda)$. We set $m_{\sigma}=(m_{\Lambda_{\sigma}},l_{\Lambda_{\sigma}})$.

As before, when $\psi|_{\Lambda}\not=0$, we can only restrict $D_{\psi}$ to $V(\Lambda)$ up to rational equivalence. In this case we just apply the previous proposition to the function $\psi-m_{\Lambda}-l_{\Lambda}$.

We end this section discussing the inverse image of a $\mathbb{T}$-Cartier divisor by an equivariant morphisms. With the notation of Proposition 4.72, let $\psi$ be an H-lattice function on $\Pi_{2}$, and $\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi_{2}}$ a set of defining vectors of $\psi$. For each $\Gamma\in\Pi_{1}$ we choose a polyhedron $\Gamma^{\prime}\in\Pi_{2}$ such that $A(\Gamma)\subset\Gamma^{\prime}$. We set $m_{\Gamma}=H^{\vee}(m_{\Gamma^{\prime}})$ and $l_{\Gamma}=m_{\Gamma^{\prime}}({\operatorname{val}}(p))+l_{\Gamma^{\prime}}$. The following proposition follows easily.