be extended to an action $T\times X\rightarrow X$.

Let us quickly recall the classical approach to toric varieties by means of *cones* and *fans*. For proofs and details the interested reader is referred to the extensive treatments [@Danilov], [@Fulton], [@Oda] and the recent and quite comprehensive [@CLS].

As usual $M$ denotes the *group of characters* $\chi : T \to \C^*$ of $T$ and $N$ the *group of 1–parameter subgroups* $\lambda : \C^* \to T$. It follows that $M$ and $N$ are $n$–dimensional dual lattices via the pairing $$\begin{array}{ccc}
M\times N & \longrightarrow & \operatorname{Hom}(\C^*,\C^*)\cong\C^*\\
 \left( \chi,\lambda \right) & \longmapsto
& \chi\circ\lambda
\end{array}$$ which translates into the standard paring $\langle u,v\rangle=\sum u_i v_i$ under the identifications $M\cong\Z^n\cong N$ obtained by setting $\chi(\tt)=\tt^{\uu}:=\prod t_i^{u_i}$ and $\lambda(t)=t^{\v}:=(t^{v_1},\ldots,t^{v_n})$.

Cones and affine toric varieties {#ssez:TV}
--------------------------------

Define $N_{\R}:=N\otimes \R$ and $M_{\R}:=M\otimes\R\cong \operatorname{Hom}(N,\Z)\otimes\R \cong \operatorname{Hom}(N_{\R},\R)$.

A *convex polyhedral cone* (or simply a *cone*) $\sigma$ is the subset of $N_{\R}$ defined by $$\sigma = \langle \v_1,\ldots,\v_s\rangle:=\{ r_1 \v_1 + \dots + r_s \v_s \in N_{\R} \mid r_i\in\R_{\geq 0} \}$$ Vectors $\v_1,\ldots,\v_s\in N_{\R}$ are said *to generate* $\sigma$; $\v_i$ is called a *primitive* generator if it generates the semigroup $\langle\v_i\rangle\cap N$. A cone $\s=\langle \v_1,\ldots,\v_s\rangle$ is called *rational* 