A model over $S$ of $X$ is a flat scheme ${\mathcal{X}}$ of finite type over $S$ together with a fixed isomorphism $X\simeq\mathcal{X}_{\eta}$. This isomorphism is part of the model, and so we can identify ${\mathcal{X}}_{\eta}$ with $X$. When $X$ is proper, we say that the model is proper whenever the scheme ${\mathcal{X}}$ is proper over $S$.