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.

We next study the inverse image of a $\mathbb{T}$-Cartier divisor with respect to equivariant morphisms as those in Theorem 4.9. Let $N_{i}$, $\Sigma_{i}$, $i=1,2$, and let $H\colon N_{1}\to N_{2}$ and $p\in X_{\Sigma_{2},0}(K)$ be as in Theorem 4.9. Let $\varphi_{p,H}$ be the associated equivariant morphism, $\Psi$ a virtual support function on $\Sigma_{2}$ and $\{m^{\prime}_{\tau^{\prime}}\}_{\tau^{\prime}\in\Sigma_{2}}$ a set of defining vectors of $\Psi$. For each cone $\tau\in\Sigma_{1}$ we choose a cone $\tau^{\prime}\in\Sigma_{2}$ such that $H(\tau)\subset\tau^{\prime}$ and we write $m_{\tau}=H^{\vee}(m^{\prime}_{\tau^{\prime}})$. The following result follows easily from the definitions