A model over $S$ of $(X,L)$ is a triple $(\mathcal{X},\mathcal{L},e)$, where $\mathcal{X}$ is a model over $S$ of $X$, $\mathcal{L}$ is a line bundle on $\mathcal{X}$ and $e\geq 1$ is an integer, together with a fixed isomorphism $\mathcal{L}|_{X}\simeq L^{\otimes e}$. When $e=1$, the model $(\mathcal{X},\mathcal{L},1)$ will be denoted $(\mathcal{X},\mathcal{L})$ for short. A model of $(\mathcal{X},\mathcal{L},e)$ is called proper whenever ${\mathcal{X}}$ is proper.