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 $N$.