Let $K$ be a field and $\mathbb{T}\simeq\mathbb{G}_{m}^{n}$ a split torus over $K$. We alternatively denote it by $\mathbb{T}_{K}$ if we want to refer to its field of definition.

Toric varieties can be described in combinatorial terms as we recall in the sequel. Let $N=\operatorname{Hom}(\mathbb{G}_{m},\mathbb{T})\simeq\mathbb{Z}^{n}$ be the lattice of one-parameter subgroups of $\mathbb{T}$ and $M=\operatorname{Hom}(\mathbb{T},\mathbb{G}_{m})=N^{\vee}=\operatorname{Hom}(N,\mathbb{Z})$ its dual lattice of characters of $\mathbb{T}$. For a ring $R$ we set $N_{R}=N\otimes R$ and $M_{R}=M\otimes R$.

To a fan $\Sigma$ we associate a toric variety $X_{\Sigma}$ over $K$ by gluing together the affine toric varieties corresponding to the cones of the fan. For $\sigma\in\Sigma$, let $\sigma^{\vee}$ be the dual cone (Definition 3.67) and set

$$M_{\sigma}=\sigma^{\vee}\cap M=\{m\in M\mid\langle m,u\rangle\geq 0,\ \forall u\in\sigma\}$$

for the saturated semigroup of its lattice points. We consider the semigroup algebra

$$K[M_{\sigma}]=\Big\{\sum_{m\in M_{\sigma}}\alpha_{m}\chi^{m}\Big|\alpha_{m}\in K,\alpha_{m}=0\text{ for almost all }m\Big\}$$

of formal finite sums of elements of $M_{\sigma}$ with the natural ring structure. It is an integrally closed domain of Krull dimension $n$. We set $X_{\sigma}=\operatorname{Spec}(K[M_{\sigma}])$ for the associated affine toric variety. If $\tau$ is a face of $\sigma$ we have that $K[M_{\tau}]$ is a localization of $K[M_{\sigma}]$. Hence there is an inclusion of open sets

$$X_{\tau}=\operatorname{Spec}(K[M_{\tau}])\lhook\joinrel\longrightarrow X_{\sigma}=\operatorname{Spec}(K[M_{\sigma}]).$$

For $\sigma,\sigma^{\prime}\in\Sigma$, the affine toric varieties $X_{\sigma}$, $X_{\sigma^{\prime}}$ glue together through the open subset $X_{\sigma\cap\sigma^{\prime}}$ corresponding to their common face. Thus these affine varieties glue together to form the toric variety

$$X_{\Sigma}=\bigcup_{\sigma\in\Sigma}X_{\sigma}.$$

This is a normal variety over $K$ of dimension $n$. When we need to specify the field of definition we will denote it as $X_{\Sigma,K}$. We denote by $\mathcal{O}_{X_{\Sigma}}$ its structural sheaf and by $\mathcal{K}_{X_{\Sigma}}$ its sheaf of rational functions. The open subsets $X_{\sigma}\subset X_{\Sigma}$ may be denoted by $X_{\Sigma,\sigma}$ when we want to include the ambient toric variety in the notation.

The cone $\{0\}$, that we denote simply by $0$, is a face of every cone and its associated affine scheme

$$X_{0}=\operatorname{Spec}(K[M])$$

is an open subset of all of the schemes $X_{\sigma}$. This variety is an algebraic group over $K$ canonically isomorphic to $\mathbb{T}$. We identify this variety with $\mathbb{T}$ and call it the principal open subset of $X_{\Sigma}$.

For each $\sigma\in\Sigma$, the homomorphism

$$K[M_{\sigma}]\to K[M]\otimes K[M_{\sigma}],\quad\chi^{m}\mapsto\chi^{m}\otimes\chi^{m}$$

induces an action of $\mathbb{T}$ on $X_{\sigma}$. This action is compatible with the inclusion of open sets and so it extends to an action on the whole of $X_{\Sigma}$

$$\mu\colon\mathbb{T}\times X_{\Sigma}\longrightarrow X_{\Sigma}.$$

Thus we have obtained a toric variety in the sense of Definition 4.1. In fact, all toric varieties are obtained in this way.

For each $\sigma\in\Sigma$, the set of $K$-rational points in $X_{\sigma}$ can be identified with the set of semigroup homomorphisms from $(M_{\sigma},+)$ to the semigroup $(K,\times):=K^{\times}\cup\{0\}$. That is,

$$X_{\sigma}(K)=\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},(K,\times)).$$

In particular, the set of $K$-rational points of the algebraic torus can be written intrinsically as

$$\mathbb{T}(K)=\operatorname{Hom}_{\operatorname{sg}}(M_{0},(K,\times))=\operatorname{Hom}_{\text{\rm gp}}(M,K^{\times})\simeq(K^{\times})^{n}.$$

Every affine toric variety has a distinguished rational point: we will denote by $x_{\sigma}\in X_{\sigma}(K)=\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},(K,\times))$ the point given by the semigroup homomorphism

$$M_{\sigma}\ni m\longmapsto\begin{cases}1,&\text{ if }-m\in M_{\sigma},\\ 0,&\text{ otherwise}.\end{cases}$$

For instance, the point $x_{0}\in X_{0}=\mathbb{T}$ is the unit of $\mathbb{T}$.

Most algebro-geometric properties of the toric scheme translate into combinatorial properties of the fan. In particular, $X_{\Sigma}$ is proper if and only if the fan is complete in the sense that $|\Sigma|=N_{\mathbb{R}}$. The variety $X_{\Sigma}$ is smooth if and only if every cone $\sigma\in\Sigma$ can be written as $\sigma=\mathbb{R}_{\geq 0}v_{1}+\cdots+\mathbb{R}_{\geq 0}v_{k}$ with $v_{1},\dots,v_{k}$ which are part of an integral basis of $N$.

The action of the torus induces a decomposition of a toric variety into disjoint orbits. These orbits are in one to one correspondence with the cones of the fan. Let $\sigma\in\Sigma$ and set

(4.4)

$$N(\sigma)=N/(N\cap\mathbb{R}\sigma),\quad M(\sigma)=N(\sigma)^{\vee}=M\cap\sigma^{\bot},$$

where $\sigma^{\bot}$ denotes the orthogonal space to $\sigma$. We will denote by $\pi_{\sigma}\colon N\to N(\sigma)$ the projection of lattices. By abuse of notation, we will also denote by $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ the induced projection of vector spaces.

The orthogonal space $\sigma^{\bot}$ is the maximal linear space inside $\sigma^{\vee}$ and $M(\sigma)$ is the maximal subgroup sitting inside the semigroup $M_{\sigma}$. Set

$$O(\sigma)=\operatorname{Spec}(K[M(\sigma)]),$$

which is a torus over $K$ of dimension $n-\dim(\sigma)$. The surjection of rings

$$K[M_{\sigma}]\longrightarrow K[M(\sigma)],\quad\chi^{a}\longmapsto\begin{cases}\chi^{a},&\text{ if }a\in\sigma^{\bot},\\ 0,&\text{ if }a\notin\sigma^{\bot},\end{cases}$$

induces a closed immersion $O(\sigma)\hookrightarrow X_{\sigma}$. In terms of rational points, the inclusion $O(\sigma)(K)\hookrightarrow X_{\sigma}(K)$ sends a group homomorphism $\gamma\colon M(\sigma)\to K^{\times}$ to the semigroup homomorphism ${\widetilde{\gamma}}\colon M_{\sigma}\to(K,\times)$ obtained by extending $\gamma$ by zero. In particular, the distinguished point $x_{\sigma}\in X_{\sigma}(K)$ belongs to the image of $O(\sigma)(K)$ by the above inclusion. Composing with the open immersion $X_{\sigma}\hookrightarrow X_{\Sigma}$, we identify $O(\sigma)$ with a locally closed subvariety of $X_{\Sigma}$. For instance, the orbit associated to the cone $0$ agrees with the principal open subset $X_{0}$. In fact, if we consider $x_{\sigma}$ as a rational point of $X_{\Sigma}$, then $O(\sigma)$ agrees with the orbit of $x_{\sigma}$ by $\mathbb{T}$.

We denote by $V(\sigma)$ the Zariski closure of $O(\sigma)$ with its induced structure of reduced closed subvariety of $X_{\Sigma}$. The subvariety $V(\sigma)$ has a natural structure of toric variety. To see it, we consider the fan on $N(\sigma)_{\mathbb{R}}$

(4.5)

$$\Sigma(\sigma):=\{\pi_{\sigma}(\tau)|\tau\supset\sigma\}.$$

This fan is called the star of $\sigma$ in $\Sigma$. For each $\tau\in\Sigma$ with $\sigma\subset\tau$, set ${\overline{\tau}}=\pi_{\sigma}(\tau)\in\Sigma(\sigma)$. Then, $M(\sigma)_{{\overline{\tau}}}=M(\sigma)\cap M_{\tau}.$ There is a surjection of rings

$$K[M_{\tau}]\longrightarrow K[M(\sigma)_{{\overline{\tau}}}],\quad\chi^{m}\longmapsto\begin{cases}\chi^{m},&\text{ if }m\in\sigma^{\bot},\\ 0,&\text{ if }m\notin\sigma^{\bot},\end{cases}$$

that defines a closed immersion $X_{{\overline{\tau}}}\hookrightarrow X_{\tau}$. These maps glue together to give a closed immersion $\iota_{\sigma}\colon X_{\Sigma(\sigma)}\hookrightarrow X_{\Sigma}$.

In view of this proposition, we will identify $V(\sigma)$ with $X_{\Sigma(\sigma)}$ and consider it a toric variety.

We now discuss more general equivariant morphisms of toric varieties.

Toric morphisms are equivariant. Indeed, a morphism is toric if and only if it is equivariant and sends the distinguished point $x_{1,0}\in X_{1}(K)$ to the distinguished point $x_{2,0}\in X_{2}(K)$.

The inclusion $V(\sigma)\to X_{\Sigma}$ is an example of equivariant morphism that is not toric. Moreover, the underlying morphism of tori depends on the choice of a section of the projection $\pi_{\sigma}\colon N\to N(\sigma)$.

Equivariant morphisms whose image intersects the principal open subset can be characterized in combinatorial terms. Let $\mathbb{T}_{i}$, $i=1,2$, be split tori over $K$. Put $N_{i}=\operatorname{Hom}(\mathbb{G}_{m},\mathbb{T}_{i})$ and let $\Sigma_{i}$ be fans in $N_{i,\mathbb{R}}$. Let $H\colon N_{1}\to N_{2}$ be a linear map such that, for every cone $\sigma_{1}\in\Sigma_{1}$, there exists a cone $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$, and let $p\in X_{\Sigma_{2},0}(K)$ be a rational point. The linear map induces a group homomorphism

$$\rho_{H}\colon\mathbb{T}_{1}\to\mathbb{T}_{2}.$$

Let $\sigma_{i}\in\Sigma_{i}$, $i=1,2,$ be cones such that $H(\sigma_{1})\subset\sigma_{2}$. Let $H^{\vee}\colon M_{2}\to M_{1}$ be the map dual to $H$. Then there is a homomorphism of semigroups $M_{2,\sigma_{2}}\to M_{1,\sigma_{1}}$ which we also denote by $H^{\vee}$. For a monomial $\chi^{m}\in K[M_{2,\sigma_{2}}]$ we denote by $\chi^{H^{\vee}m}$ its image in $K[{M_{1,\sigma_{1}}}]$. The assignment $\chi^{m}\mapsto\chi^{m}(p)\chi^{H^{\vee}m}$ induces morphisms of algebras $K[M_{2,\sigma_{2}}]\to K[M_{1,\sigma_{1}}],$ that, in turn, induce morphisms

$$X_{\sigma_{1}}=\operatorname{Spec}(K[M_{1,\sigma_{1}}])\longrightarrow X_{\sigma_{2}}=\operatorname{Spec}(K[M_{2,\sigma_{2}}]).$$

These morphisms are compatible with the restriction to open subsets, and they glue together into a $\rho_{H}$-equivariant morphism

(4.8)

$$\varphi_{p,H}\colon X_{\Sigma_{1}}\longrightarrow X_{\Sigma_{2}}.$$

In case $p=x_{2,0}$, the distinguished point on the principal open subset of $X_{\Sigma_{2}}$, this morphism is a toric morphism and will be denoted as $\varphi_{H}$ for short.

General equivariant morphisms are obtained composing an equivariant morphism of the form $\varphi_{p,H}\colon X_{\Sigma_{1}}\rightarrow X_{\Sigma_{2}}$ with the inclusion of $X_{\Sigma_{2}}$ as a toric orbit of a third toric variety.

We now show how to refine the Stein factorization for an equivariant morphism in terms of the combinatorial data. Let $N_{i}$, $\Sigma_{i}$ $H$ and $p$ be as in Theorem 4.9. The linear map $H$ factorizes as

$$N_{1}\overset{H_{\operatorname{surj}}}{\relbar\joinrel\twoheadrightarrow}N_{3}:=H(N_{1})\overset{H_{\operatorname{sat}}}{\lhook\joinrel\longrightarrow}N_{4}:=\operatorname{sat}(N_{3})\overset{H_{\operatorname{inj}}}{\lhook\joinrel\longrightarrow}N_{2},$$

where $N_{3}$ is the image of $H$ and $N_{4}$ is the saturation of $N_{3}$ with respect to $N_{2}$. Clearly $N_{3,\mathbb{R}}=N_{4,\mathbb{R}}$. By restriction, the fan $\Sigma_{2}$ induces a fan in this linear space. We will call this fan either $\Sigma_{3}$ or $\Sigma_{4}$, depending on the lattice we are considering. Applying the combinatorial construction of equivariant morphisms, we obtain a diagram

$$X_{\Sigma_{1}}\overset{\varphi_{H_{\operatorname{surj}}}}{\longrightarrow}X_{\Sigma_{3}}\overset{\varphi_{H_{\operatorname{sat}}}}{\longrightarrow}X_{\Sigma_{4}}\overset{\varphi_{p,H_{\operatorname{inj}}}}{\longrightarrow}X_{\Sigma_{2}},$$

where the first morphism has connected fibres (see [Oda88, Proposition 1.14]), the second morphism is finite and surjective.

The third morphism is also finite and can be further factorized as a normalization followed by a closed immersion. In general, consider a saturated sublattice $Q$ of $N$, $\Sigma$ a fan in $N_{\mathbb{R}}$ and $p\in X_{\Sigma,0}(K)$. Let $\Sigma_{Q}$ be the induced fan in $Q_{\mathbb{R}}$ and $\iota\colon Q\hookrightarrow N$ the inclusion of $Q$ into $N$. Then, we have a finite equivariant morphism

$$\varphi_{p,\iota}\colon X_{\Sigma_{Q}}\longrightarrow X_{\Sigma}.$$

Set $P=Q^{\vee}=M/Q^{\bot}$ and let $\iota^{\vee}\colon M\to P$ be the dual of $\iota$. Let $\sigma\in\Sigma$ and $\sigma^{\prime}=\sigma\cap Q_{\mathbb{R}}\in\Sigma_{Q}$. The natural semigroup homomorphisms $M_{\sigma}\to P_{\sigma^{\prime}}$ factors as

$$M_{\sigma}\relbar\joinrel\twoheadrightarrow{M_{Q,\sigma}}:=(M_{\sigma}+Q^{\bot})/Q^{\bot}\lhook\joinrel\longrightarrow P_{\sigma^{\prime}}:=P\cap(\sigma^{\prime})^{\vee}.$$

The first arrow is the projection and will be denoted as $m\mapsto[m]$, while the second one is the inclusion of ${M_{Q,\sigma}}$ into its saturation with respect to $P$. We have a diagram of $K$-algebra morphisms

$$K[M_{\sigma}]\relbar\joinrel\twoheadrightarrow K[{M_{Q,\sigma}}]\lhook\joinrel\longrightarrow K[P_{\sigma^{\prime}}],$$

where the left map is given by $\chi^{m}\mapsto\chi^{m}(p)\chi^{[m]}$, and the right map is given by $\chi^{[m]}\mapsto\chi^{\iota^{\vee}m}$. Let $Y_{\sigma,Q,p}\simeq\operatorname{Spec}(K[{M_{Q,\sigma}}])$ be the closed subvariety of $X_{\sigma}$ given by the left surjection. Then we have induced maps

$$X_{\sigma^{\prime}}\relbar\joinrel\twoheadrightarrow Y_{\sigma,Q,p}\lhook\joinrel\longrightarrow X_{\sigma}.$$

These maps are compatible with the restriction to open subsets and so they glue together into maps

(4.11)

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

Then $Y_{\Sigma,Q,p}$ is the closure of the orbit of $p$ under the action of the subtorus of $\mathbb{T}$ determined by $Q$, while the toric variety $X_{\Sigma_{Q}}$ is the normalization of $Y_{\Sigma,Q,p}$.

When $p=x_{0}$, the subvariety $Y_{\Sigma,Q,p}$ will be denoted by $Y_{\Sigma,Q}$ for short.

A translated toric subvariety is not necessarily a toric variety in the sense of Definition 4.1, since it may be non-normal.

When studying toric varieties, the objects that admit a combinatorial description are those that are compatible with the torus action. These objects are enough for many purposes. For instance, the divisor class group of a toric variety is generated by invariant divisors.

Let $\pi_{2}\colon\mathbb{T}\times X\to X$ denote the projection to the second factor and $\mu\colon\mathbb{T}\times X\to X$ the torus action. A Cartier divisor $D$ is invariant if and only if

$$\pi_{2}^{\ast}D=\mu^{\ast}D.$$

The combinatorial description of $\mathbb{T}$-Cartier divisors is done in terms of virtual support functions.

A support function on a complete fan in the sense of the previous definition, is the support function of a polytope as in Example 3.16: it is the support function of the polytope

$$\operatorname{conv}(\{m_{\sigma}\}_{\sigma\in\Sigma^{n}})\subset M_{\mathbb{R}},$$

where $\Sigma^{n}$ is the subset of $n$-dimensional cones of $\Sigma$.

Two vectors $m,m^{\prime}\in M$ define the same functional on a cone $\sigma$ if and only if $m-m^{\prime}\in\sigma^{\bot}$. Hence, for a given virtual support function $\Psi$ on a fan $\Sigma$, each defining vector $m_{\sigma}$ is unique up to the orthogonal space $\sigma^{\bot}$. In particular, $m_{\sigma}\in M$ is uniquely defined for $\sigma\in\Sigma^{n}$ and, in the other extreme, $m_{0}$ can be any point of $M$.

Let $\{m_{\sigma}\}_{\sigma\in\Sigma}$ be a set of defining vectors of $\Psi$. These vectors have to satisfy the compatibility condition

(4.16)

$$m_{\sigma}|_{\sigma\cap\sigma^{\prime}}=m_{\sigma^{\prime}}|_{\sigma\cap\sigma^{\prime}}\mbox{ for all }\sigma,\sigma^{\prime}\in\Sigma.$$

On each open set $X_{\sigma}$, the vector $m_{\sigma}$ determines a rational function $\chi^{-m_{\sigma}}$. For $\sigma,\sigma^{\prime}\in\Sigma$, the above compatibility condition implies that $\chi^{-m_{\sigma}}/\chi^{-m_{\sigma^{\prime}}}$ is a regular function on the overlap $X_{\sigma}\cap X_{\sigma^{\prime}}=X_{\sigma\cap\sigma^{\prime}}$ and so $\Psi$ determines a Cartier divisor on $X_{\Sigma}$:

(4.17)

$$D_{\Psi}:=\left\{(X_{\sigma},\chi^{-m_{\sigma}})\right\}_{\sigma\in\Sigma}.$$

This Cartier divisor does not depend on the choice of defining vectors and it is a $\mathbb{T}$-Cartier divisor. All $\mathbb{T}$-Cartier divisors are obtained in this way.

We next recall the relationship between Cartier divisors and line bundles in the toric case.

To a Cartier divisor $D$, one associates an invertible sheaf of fractional ideals of $\mathcal{K}_{X}$, denoted $\mathcal{O}(D)$. When $D$ is a $\mathbb{T}$-Cartier divisor given by a set of defining vectors, $\{m_{\sigma}\}_{\sigma\in\Sigma}$, the sheaf $\mathcal{O}(D)$ can be realized as the subsheaf of $\mathcal{O}_{X}$-modules generated, in each open subset $X_{\sigma}$, by the rational function $\chi^{m_{\sigma}}$. The section $1\in\mathcal{K}_{X}$ provides us with a distinguished rational section $s_{D}$ such that $\operatorname{div}(s_{D})=D$. Since $D$ is supported on the complement of the principal open subset, $s_{D}$ is regular and no-where vanishing on $X_{0}$. We set $z=s_{D}(x_{0})$. This is a toric structure on $\mathcal{O}(D)$. From now on, we will assume that $\mathcal{O}(D)$ is equipped with this toric structure. Then $((\mathcal{O}(D),z),s_{D})$ is a toric line bundle with a toric section.

Thanks to this result and Theorem 4.18, we can freely move between the languages of virtual support functions, $\mathbb{T}$-Cartier divisors, and toric line bundles with a toric section.

We next recall the relationship between Cartier divisors and Weil divisors in the toric case.

The invariant hypersurfaces of a toric variety are particular cases of the toric subvarieties considered in the previous section: they are the varieties of the form $V(\tau)$ for $\tau\in\Sigma^{1}$. Hence, a $\mathbb{T}$-Weil divisor is a finite formal linear combination of subvarieties of the form $V(\tau)$ for $\tau\in\Sigma^{1}$.

Since the toric variety $X$ is normal, each Cartier divisor determines a Weil divisor. This correspondence associates to the $\mathbb{T}$-Cartier divisor $D_{\Psi}$, the $\mathbb{T}$-Weil divisor

(4.25)

$$[D_{\Psi}]=\sum_{\tau\in\Sigma^{1}}-\Psi(v_{\tau})V(\tau),$$

where $v_{\tau}\in N$ is the smallest nonzero lattice point in $\tau$.

For a toric variety $X_{\Sigma}$ of dimension $n$, we denote by $\operatorname{Div}_{\mathbb{T}}(X_{\Sigma})$ its group of $\mathbb{T}$-Cartier divisors, and by $Z_{n-1}^{\mathbb{T}}(X_{\Sigma})$ its group of $\mathbb{T}$-Weil divisors. Recall that $\operatorname{Pic}(X_{\Sigma})$, the Picard group of $X_{\Sigma}$, is the group of isomorphism classes of line bundles. Let $A_{n-1}(X_{\Sigma})$ denote the Chow group of cycles of dimension $n-1$. The following result shows that these groups can computed in terms of invariant divisors.

In view of Theorem 4.22, the upper exact sequence of the diagram in Theorem 4.27 can be interpreted as follows.

We next study the intersection of a $\mathbb{T}$-Cartier divisor with the closure of an orbit. Let $\Sigma$ be a fan in $N_{\mathbb{R}}$ and $\Psi$ the virtual support function on $\Sigma$ given by the set of defining vectors $\{m_{\tau}\}_{\tau\in\Sigma}$. Let $\sigma$ be a cone of $\Sigma$ and $\iota_{\sigma}\colon V(\sigma)\hookrightarrow X_{\Sigma}$ the associated closed immersion. We consider first the case when $\Psi|_{\sigma}=0$. Let $\tau\supset\sigma$ be another cone of $\Sigma$. For vectors $u\in\tau$ and $v\in\mathbb{R}\sigma$ such that $u+v\in\tau$, the condition $\Psi|_{\sigma}=0$ implies

$$\Psi(u+v)=\langle m_{\tau},u+v\rangle=\langle m_{\tau},u\rangle=\Psi(u)$$

because $m_{\tau}\big|_{\mathbb{R}\sigma}=0$. Hence, we can define a function

(4.30)

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

for any $v\in\mathbb{R}\sigma$ such that $u+v\in\bigcup_{\tau\supset\sigma}\tau$.

It is easy to produce a set of defining vectors of $\Psi(\sigma)$. For each cone $\tau\supset\sigma$ we denote by ${\overline{\tau}}=\pi_{\sigma}(\tau)$ the corresponding cone in $\Sigma(\sigma)$. Since $m_{\tau}\big|_{\mathbb{R}\sigma}=0$, then $m_{\tau}\in M(\sigma)=M\cap\sigma^{\perp}$. We set $m_{{\overline{\tau}}}=m_{\tau}\in M(\sigma)$.

When $\Psi|_{\sigma}\not=0$, the cycles $D_{\Psi}$ and $V(\sigma)$ do not intersect properly, and we can only intersect $D_{\Psi}$ with $V(\sigma)$ up to rational equivalence. To this end, we choose any $m_{\sigma}^{\prime}$ such that $\Psi(u)=\langle m^{\prime}_{\sigma},u\rangle$ for every $u\in\sigma$. Then the divisor $D_{\Psi-m_{\sigma}^{\prime}}$ is rationally equivalent to $D_{\Psi}$ and $\Psi-m_{\sigma}^{\prime}|_{\sigma}=0$. By the above result, this divisor intersects $V(\sigma)$ properly, and its restriction to $V(\tau)$ is given by the virtual support function $(\Psi-m_{\sigma}^{\prime})(\sigma)$.

Let now $(L,z)$ be a toric line bundle on $X_{\Sigma}$ and $\sigma\in\Sigma$. The line bundle $\iota^{\ast}_{\sigma}L$ on $V(\sigma)$ has an induced toric structure. Let $s$ be a toric section of $L$ that is regular and nowhere vanishing on $X_{\sigma}$, and set $z_{\sigma}=s(x_{\sigma})\in L_{x_{\sigma}}\setminus\{0\}$. If $s^{\prime}$ is another such section, then $s^{\prime}=\chi^{m}s$ for an $m\in M$ such that $m|_{\sigma}=0$, by Corollary 4.29. Therefore $s^{\prime}(x_{\sigma})=s(x_{\sigma})$. Hence, $z_{\sigma}$ does not depend on the choice of section and $(\iota^{\ast}_{\sigma}L,z_{\sigma})$ is the induced toric line bundle. The following result follows easily from the constructions.