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.