lattice polytope.

\[prop:semiampio\] Let $|\Si|$ be convex of full dimension. Then $D$ is semi-ample if and only if $kD$ is numerically effective (nef), for some $k\in\N$, that is $kD$ is Cartier and $kD\cdot C\geq 0$, for any complete curve $C\subset X$.

Starting from a lattice polytope $\D$ one can construct a *projective* toric variety $\P_\D$ as follows. For any nonempty face $\phi< \D$ consider the dual cone $\s_\phi^\vee\subseteq N_\R$ of the cone $$\s_\phi:=\{r(\m-\m')\ |\ \m\in\D\ ,\ \m'\in \phi\ ,\ r\in\R_{\geq 0}\}\subseteq M_{\R}$$ Then $\Si_{\D}^\perp:=\{\s_\phi^\vee\ |\ \phi< \D\}$ turns out to be a fan, called the *normal fan* of the polytope $\D$, and $\P_\D$ is the associated toric variety. It is projective as there exists an ample divisor $H$ of $\P_{\D}$ whose associated polytope is precisely $\D$.

A further *complete* toric variety $\XX_\D$ can be associated with a lattice polytope $\D$ such that $\0\in\operatorname{Int}\D$. Namely, for every facet $\Phi<\D$ consider the cone *projecting $\Phi$ from the origin*, that is $$\label{cono su facciata}
  \s_\Phi:=\{r\m\,|\,\m\in\Phi\ ,\ r\in\R_{\geq 0}\}\subseteq M_\R$$ Then $\Si_\D:=\{\tau\,|\,\exists\,\Phi<^1\D:\ \tau<\s_\Phi\}$ turns out to be a fan, called the *fan over the polytope $\D$*, and $\XX_\D$ is the associated toric variety. It is complete as the support $|\Si_\D|$ is the whole $M_\R$ 