Consider a model ${{\mathscr{X}}}$ of the proper variety $X$ over $K$. The rational vector space space $N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ is by definition the quotient of ${{\rm Pic}\,}({{\mathscr{X}}})_{\mathbb{Q}}\coloneqq{\rm Pic}({{\mathscr{X}}})\otimes_{\mathbb{Z}}\mathbb{Q}$ by the subspace generated by classes of line bundles ${\mathscr{L}}$ such that ${\mathscr{L}}\cdot C=0$ for each closed curve $C$ in the special fiber ${{\mathscr{X}}}_{s}$. Note that $N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ is finite dimensional by applying [Kle66, Prop. IV.1.4] to ${{\mathscr{X}}}_{s}$. We define $N^{1}({{\mathscr{X}}}/S)\coloneqq N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{R}$. An element $\alpha\in N^{1}({{\mathscr{X}}}/S)_{\mathbb{Q}}$ (resp. $\alpha\in N^{1}({{\mathscr{X}}}/S))$ is called nef if $\alpha\cdot C\geq 0$ for all closed curves $C$ in ${{\mathscr{X}}}_{s}$. We call a line bundle ${\mathscr{L}}$ on ${{\mathscr{X}}}$ nef if the class of ${\mathscr{L}}$ in $N^{1}({{\mathscr{X}}}/S)$ is nef.