If $Y$ is a proper variety over an arbitrary field $k$, we denote by $N^{1}(Y)_{\mathbb{Q}}$ the rational vector space ${{\rm Pic}\,}(Y)\otimes\mathbb{Q}$ modulo numerical equivalence. Similarly, we denote by $N^{1}(Y)=N^{1}(Y)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{R}$ the real vector space ${{\rm Pic}\,}(Y)\otimes\mathbb{R}$ modulo numerical equivalence. A class in $N^{1}(Y)$ is called ample if it is an $\mathbb{R}_{>0}$-linear combination of classes induced by ample line bundles on $Y$. An element $\alpha\in N^{1}(Y)_{\mathbb{Q}}$ (resp. $\alpha\in N^{1}(Y))$ is called nef if $\alpha\cdot C\geq 0$ for all closed curves $C$ in $Y$.