(2.1.3) Let $X$ be a smooth and proper scheme over $C$ with geometrically connected fibers. A model of $X$ over $\mathscr{C}$ is a flat separated $\mathscr{C}$-scheme of finite type $\mathscr{X}$ endowed with an isomorphism of $C$-schemes $\mathscr{X}\times_{\mathscr{C}}C\to X$. Note that we do not require $\mathscr{X}$ to be proper over $\mathscr{C}$. Morphisms of models are defined in the usual way. We denote by $\mathscr{X}_{s}$ the fiber of $\mathscr{X}$ over $s$, by $\mathscr{X}_{R}$ the base change of $\mathscr{X}$ to $\mathrm{Spec}\,R$ and by $X_{K}$ the base change of $X$ to $\mathrm{Spec}\,K$. We denote by $K_{X}$ a relative canonical divisor for $X$ over $C$, and for every normal model $\mathscr{X}$ of $X$, we denote by $K_{\mathscr{X}}$ a relative canonical divisor for $\mathscr{X}$ over $\mathscr{C}$.