Let $T$ be a torus which we fix. Recall that there was a correspondence between fans on $X_\ast(T)$ and normal toric varieties for $T$.
Divisors and line bundles
Proposition 1. There is a bijection between faces in a fan $\Sigma$ and $T$-orbits of $X(\Sigma)$, where $\tau \in \Sigma$ corresponds to $\Spec k[\tau^\perp \cap X^\ast(T)]$.
We can check that the dimension of this orbit is $\dim O_\tau = n - \dim \tau$. So $T$-stable codimension $1$ subvarieties correspond to edges of $\Sigma$. Given any $u \in X^\ast(T)$, this defines a function on $T$ and hence a rational function on $X(\Sigma)$. When $D_i = \operatorname{cone}(v_i)$, we can check that $$ \operatorname{ord}_ {D_i} \chi^u = \langle u, v_i \rangle. $$ So we can write $$ \operatorname{div}(\chi^u) = \sum_i \langle u, v_i \rangle D_i, $$ and then define $$ \operatorname{Pic}_ T(X(\Sigma)) = \operatorname{Weil}_ T(X(\Sigma)) / \mathrm{prin}. $$ This turns out to be the actual Picard group.
Proposition 2. The $\chi^u$ for $u \in X^\ast(T)$ satisfying $\langle u, v_i \rangle + a_i \ge 0$, where $D = \sum a_i D_i$, span the vector space $H^0(X(\Sigma), \mathscr{O}(D))$.
To such a $T$-divisor $D = \sum a_i D_i$, we can associate a polytope $$ P_D = \lbrace u \in X^\ast(T)_ \mathbb{R} : \langle u, v_i \rangle + a_i \ge 0 \rbrace. $$ Then we can also interpret $H^0(X, \mathscr{O}(D))$ as having a basis corresponding to elements of $P_D \cap X^\ast(T)$.
Remark 3. Giving a divisor is also equivalent to giving a piecewise-linear function $\psi_D \colon \operatorname{supp}(\Sigma) \to \mathbb{R}$ with the property that for all $\sigma \in \Sigma$ there exists an element $\alpha \in X^\ast(T)$ for which $\psi \vert_\sigma = \alpha \vert_\sigma$. Some things are more natural in this setting. For example, if $\operatorname{supp}(\Sigma) = X_\ast(T)_ \mathbb{R}$ and $\psi_D$ is convex, then $\mathscr{O}(D)$ is globally generated.
From polytopes to toric varieties
Instead of starting with a fan, we can start with a convex lattice polytope $\Delta \subseteq X^\ast(T)_ \mathbb{R}$. For a point $\tau \in \Delta$, we can define the cone $$ C_\tau(\Delta) = \langle u - \tau : u \in \Delta \rangle. $$ Collecting the dual of these things, we obtain $$ \Sigma(\Delta) = \lbrace C_\tau^\vee(\Delta) : \tau \in \Delta \rbrace, \quad \operatorname{Tor}(\Delta) = X(\Sigma(\Delta)). $$ On this thing, there is a natural divisor $$ D = -\sum_i \psi(v_i) D_i, \quad \psi(v) = \min_{u \in \Delta \cap X^\ast(T)} \langle u, v \rangle. $$ Then we will have $P_D = \Delta$.
By construction, this is globally generated, so there is a map $$ \varphi \colon \operatorname{Tor}(\Delta) \to \mathbb{P}^{N-1}, \quad N = \lvert \Delta \cap X^\ast(T) \rvert. $$ When $n = 2$, this is an embedding. For $n \ge 3$, there needs to be some condition on $\Delta$ for this to be an embedding.
Moment maps
Let us now work over $\mathbb{C}$. We can identify $\mathbb{C}^\times \cong S^1 \times \mathbb{R}$. Then $T(\mathbb{C}) \cong (S^1)^n \times \mathbb{R}^n$. Now there is a map $$ \mu \colon \operatorname{Tor}(\Delta) \to X^\ast(T)_ \mathbb{R}; \quad x \mapsto \frac{\sum_{u \in \Delta \cap X^\ast(T)} \lvert \chi^u(x) \rvert u}{\sum_{u \in \Delta \cap X^\ast(T)} \lvert \chi^u(x) \rvert} $$ where we regard $\chi^u$ as global sections on the line bundle $\mathscr{O}(D)$ from above.
Proposition 4. This map $\mu$ has image exactly $\Delta \subseteq X^\ast(T)_ \mathbb{R}$. Moreover, it is invariant under the $(S^1)^n$-action, hence descends to a map $$ \bar{\mu} \colon \operatorname{Tor}(\Delta)_ \ge = \operatorname{Tor}(\Delta) / (S^1)^n \to \Delta. $$
Example 5. In the case of $\Delta$ being the triangle with vertices $(0,0)$ and $(0,1)$ and $(1,0)$, we see that $\operatorname{Tor}(\Delta) = \mathbb{P}^2$. Then we see that $$ \mu([x : y : z]) = \frac{(\lvert y \rvert, \lvert z \rvert)}{\lvert x \rvert + \lvert y \rvert + \lvert z \rvert}. $$ This has image exactly $\Delta$, and the preimages of faces correspond to the $T$-orbits.
Proposition 6. This map $\bar{\mu} \colon \operatorname{Tor}(\Delta) / (S^1)^n \to \Delta$ is a homeomorphism and for every face $Q \subseteq \Delta$ corresponding to $\sigma$ a cone in $\Sigma(\Delta)$, the map $\bar{\mu}$ maps $O_\sigma$ bijectively onto the interior $\operatorname{Int}(Q)$.
Let us now work over $\mathbb{R}$. Here we still have a map $$ \operatorname{Tor}_ \mathbb{R}(\Delta) \to \mathbb{RP}^{N-1}. $$ On the other hand, we can consider $$ T(\mathbb{R}) = (\mathbb{R}^\times)^n = \coprod_{\epsilon \in \lbrace \pm 1 \rbrace^n} (\mathbb{R}^n)_ \epsilon, \quad \operatorname{Tor}_ {\mathbb{R},\epsilon}(\Delta) = \overline{(\mathbb{R}^n)_ \epsilon} \subseteq \operatorname{Tor}_ \mathbb{R}(\Delta). $$
Assume that $\Delta \in \mathbb{R}_ {\ge 0}^n$ and then consider the reflections $\Delta_\epsilon = r_\epsilon(\Delta)$, where $r_\epsilon \colon \mathbb{R}_ {\gt 0}^n \to (\mathbb{R}^n)_ \epsilon$ is the reflection. Then we obtain $$ \mu_{\Delta,\epsilon} = r_\epsilon \circ \bar{\mu} \colon \operatorname{Tor}_ {\mathbb{R},\epsilon}(\Delta) \to \Delta_ \epsilon. $$
We also have a complexified moment map $$ \mathbb{C}\mu \colon T = \operatorname{Int}(\operatorname{Tor}(\Delta)) \to \mathbb{C}\Delta = \Delta \times (S^1)^n; \quad x \mapsto \frac{\sum_{u \in \Delta \cap X^\ast(T)} \chi^u(x) u}{\sum_{u \in \Delta \cap X^\ast(T)} \lvert \chi^u(x) \rvert}. $$
Hypersurfaces
Proposition 7. There is a bijection between
- hyperplane sections $S \subseteq \operatorname{Tor}(\Delta)$ (with respect to $\varphi \colon \operatorname{Tor}(\Delta) \to \mathbb{P}^{N-1}$) not containing any boundary divisor $D_i$,
- functions $f \colon T \to k$ given by $f(t) = \sum_{i \in \Delta \cap X^\ast(T)} a_i t^i$ containing at least two monomials,
where $f$ corresponds to the closure of $\lbrace f = 0 \rbrace$.
This closure also can be described quite explicitly.