Throughout this section, let $Z$ be an $r$-dimensional (normal) proper toric variety over $k$, in the sense of [KKMSD73]. This means that $Z$ is a normal $k$-variety, containing the torus $\mathbb{T}=\mathbb{G}^{r}_{m,k}$ as an open subset, and such that the torus action onto itself extends to an action on $Z$. The complement $\Delta_{Z}=Z\setminus\mathbb{T}$ is a reduced anticanonical Weil divisor in $Z$, called the toric boundary of $Z$; we write it as the sum of its irreducible components $\Delta_{Z}=\sum_{l\in L}Z_{l}$. We write $N=\Hom(\mathbb{G}_{m,k},\mathbb{T})$ for the free abelian group of 1-parameter subgroups of $\mathbb{T}$, and $N_{\mathbb{R}}=N\otimes\mathbb{R}$.

The variety $Z$ can be described by a combinatorial object, called its fan $\Sigma$. The fan lives inside the finite-dimensional vector space $N_{\mathbb{R}}$; $\Sigma$ is a collection of strictly convex rational polyhedral cones $\Sigma=\{\sigma\}_{\sigma\in\Sigma}$ inside $N_{\mathbb{R}}$, stable under intersection and such that each face of a cone in $\Sigma$ is itself in $\Sigma$. The cones of $\Sigma$ are in inclusion-reversing bijection with the strata of $\Delta_{Z}$; this induces in particular a bijection between the set of rays (i.e. 1-dimensional cones) of $\Sigma$ and the irreducible components of $\Delta_{Z}$.
The fan $\Sigma$ encodes various types of algebro-geometric information about $Z$. For instance, the variety $Z$ is smooth if and only if each top-dimensional cone of $\Sigma$ is $\GL(N)$-isomorphic to the standard octant $\mathbb{R}^{r}_{\geqslant 0}\subset\mathbb{R}^{r}$.

Furthermore, in the case where $Z$ is smooth (which implies it has simple normal crossing boundary), the fan allows us to give a rather explicit description of the Picard group of $Z$. Indeed, each Cartier divisor can be moved via the torus action to a $\mathbb{T}$-invariant Cartier divisor, which is thus supported on the boundary. This provides a canonical set of generators of $\Pic(Z)$, and the kernel can be described as follows.
Write $\Div^{\mathbb{T}}(Z)=\oplus_{l\in L}\mathbb{Z}Z_{l}\simeq\mathbb{Z}^{L}$ the abelian group of Weil divisors supported on the boundary. The canonical map $q:\Div^{\mathbb{T}}(Z)\longrightarrow\Pic(Z)$ sends a divisor to its class; the map $p:M\longrightarrow\Div^{\mathbb{T}}(Z)$ sends a monomial $z^{m}$ to the principal divisor $\textrm{div}(z^{m})$.

We now want to describe how the fan $\Sigma$ encodes the intersection theory on $Z$. Each $1$-cycle in $Z$ being numerically equivalent to a sum of toric strata, it is enough to study the intersection numbers $(C\cdot Z_{l})$, where $Z_{l}$ is a boundary component of $Z$ and $C$ is a 1-dimensional toric stratum, which is isomorphic to $\mathbb{P}^{1}$ by properness. The stratum $C$ is thus a rational curve with two marked points $p$ and $q$, which are the intersection points of $C$ with two components of $\Delta_{Z}$, denoted here by $Z_{p}$ and $Z_{q}$, with corresponding rays $\rho_{p}$ and $\rho_{q}$. The curve $C$ corresponds to a $(r-1)$-dimensional cone $\sigma_{C}$ of $\Sigma$, while the points $p$ and $q$ correspond to the maximal cones generated by $<\sigma_{C},\rho_{p}>$ and $<\sigma_{C},\rho_{q}>$.

Note that this lemma holds even if $Z$ is not proper.
Finally, we have the following vanishing theorem for nef divisors on a proper toric variety.

Following [NXY19], we will say that an $R$-scheme of finite type $\mathscr{Z}$ is toric if there exists a toric $k$-scheme of finite type $\mathcal{Z}$, together with a toric morphism $t:\mathcal{Z}\longrightarrow\mathbb{A}^{1}_{k}$, such that $\mathscr{Z}\simeq\mathcal{Z}\times_{\mathbb{A}^{1}}R$. Writing $\widehat{N}$ for the lattice of 1-parameter subgroups of the torus of $\mathcal{Z}$, such a scheme is described by a fan $\widehat{\Sigma}$ in $\widehat{N}_{\mathbb{R}}$, together with a linear map $\ord(t):\lvert\widehat{\Sigma}\rvert\longrightarrow\mathbb{R}_{\geq 0}$, defined by $\ord(t)(n)=\ord_{0}(t\circ n)$ for a 1-parameter subgroup $n:\mathbb{G}_{m}\rightarrow\mathcal{Z}$. Note that the map $\ord(t)$ recovers the function $t$ uniquely, since it is a monomial.