if $\v_1,\ldots,\v_s\in N$, *simplicial* if $\v_1,\ldots,\v_s$ are $\R$–linear independent and *non-singular* if primitive generators $\v_1,\ldots,\v_s$ can be extended to giving a basis of the lattice $N$.

A cone $\s$ is called *strongly convex* or *pointed* if it does not contain a linear subspace of positive dimension of $N_{\R}$.

The *dual cone $\s^{\vee}$ of $\s$* is the subset of $M_{\R}$ defined by $$\sigma^{\vee} = \{ \uu \in M_{\R} \mid \forall\ \v \in \sigma \quad \langle \uu, \v \rangle \ge 0 \}$$ A *face $\tau$ of $\s$* (denoted by $\tau <\s$) is the subset defined by $$\tau = \sigma \cap \uu^{\bot} = \{\v \in \sigma \mid \langle \uu, \v \rangle = 0 \}$$ for some $\uu\in \sigma ^{\vee}$. Observe that also $\tau$ is a cone.

A *facet* $\tau$ of a cone $\s$ is a codimension 1 face, denoted by $\tau<^1\s$.

Gordon’s Lemma ensures that the semigroup $S_{\s}:=\s^{\vee}\cap M$ is *finitely generated*. Then also the associated $\C$–algebra $A_{\s}:=\C[S_{\s}]$ is finitely generated. A choice of $m$ generators gives a presentation of $A_{\s}$ $$A_{\s}\cong \C[X_1,\dots,X_m]/I_{\s}$$ Then $U_{\s}:=\operatorname{Spec}(A_{\s})\subset\C^m$ is an *affine toric variety*. Since a closed point $x\in U_{\s}$ is an evaluation of elements in $\C[S_{\s}]$ satisfying the relations generating $I_{\s}$, then it can be identified 