Theorem 4.54 . [02QX]

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Theorem 4.54.

Let $X_{\Sigma}$ be a proper toric variety and $D_{\Psi}$ a $\mathbb{T}$ -Cartier divisor on $X_{\Sigma}$ .

(1)
The following properties are equivalent:
1. (a)
  
  $D_{\Psi}$ is ample;
2. (b)
  
  $(D_{\Psi}\cdot C)>0$ for every curve $C$ in $X_{\Sigma}$ ;
3. (c)
  
  $(D_{\Psi}\cdot V(\tau))>0$ for every $\tau\in\Sigma^{n-1}$ .
(2)
The following properties are equivalent:
1. (a)
  
  $D_{\Psi}$ is generated by its global sections;
2. (b)
  
  $(D_{\Psi}\cdot C)\geq 0$ for every curve $C$ in $X_{\Sigma}$ ;
3. (c)
  
  $(D_{\Psi}\cdot V(\tau))\geq 0$ for every $\tau\in\Sigma^{n-1}$ .