$X(\Si)$ is projective.

Polytopes of divisors and associated fans and varieties {#ssez:politopi}
-------------------------------------------------------

A *polytope* $\D\subset M_{\R}$ is the convex hull of a finite set $S$ of points, that is $\D=\operatorname{Conv}(S)$.

If $S\subseteq M$ then $\D$ is called a *lattice polytope*. When $\D$ is a full dimensional polytope its presentation as an intersection of closed half-spaces has an especially nice form, because each facet $\Phi<^1\D$ has a unique supporting affine hyperplane. We denote such an hyperplane and the corresponding closed half-space as $$H_\Phi=\{\m\in\M_\R\,|\,\langle\m,\n_\phi\rangle=-a_\Phi\}\quad,\quad H^+_\Phi=\{\m\in\M_\R\,|\,\langle\m,\n_\phi\rangle\geq-a_\Phi\}$$ where $(\n_\Phi,a_\Phi)\in N_\R\times \R$ is unique up to multiplication by a positive real number. We call $\n_\Phi$ an *inward pointing normal* vector of the facet $\Phi$. It follows that $$\label{ss-intersezione}
  \D=\bigcap_{\Phi<^1\D} H^+_\Phi= \{\m\in M_\R\,|\,\forall\,\Phi<^1\D\quad\langle\m,\n_\Phi\rangle\geq-a_\Phi\}$$ The relative interior of $\D$ will be denoted by $\operatorname{Relint}{\D}$, or simply $\operatorname{Int}\D$ when $\D$ is full dimensional.

In the following *we will consider full dimensional polytopes only, unless otherwise advised*.

The *polar polytope* $\D^*$ of a polytope $\D\subseteq M_\R$ containing the origin $\0\in M$ as an interior point, that is $\0\in\operatorname{Int}{\D}$, is defined as follows $$\label{polare}
  \D^*:=\{\n\in N_\R\,|\,\forall\,\m\in\D\quad\langle\n,\m\rangle\geq -1\}\subseteq N_\R$$ It is a full dimensional polytope in $N_\R$ with $\0\in\operatorname{Int}{\D^*}$ and $(\D^*)^*=\D$. In particular, if $\D$ admits the presentation given in 