Let $X$ be a proper scheme over $K$. Then an algebraic ${K^{\circ}}$-model of $X$ is a proper flat scheme ${\mathscr{X}}$ over ${K^{\circ}}$ with a fixed isomorphism from the generic fiber ${\mathscr{X}}_{\eta}$ to $X$. Usually, we will identify ${\mathscr{X}}_{\eta}$ with $X$ along this fixed isomorphism.

It follows from Nagata’s embedding theorem that an algebraic ${K^{\circ}}$-model of $X$ exists. The set of isomorphism classes of algebaic ${K^{\circ}}$-models of $X$ is partially order by morphisms of ${K^{\circ}}$-models of $X$ (where by definition such a map extends the identity on $X$). A diagonal argument shows easily that the set of isomorphism classes is directed with respect to this partial order.

Let $L$ be a line bundle on $X$. An algebraic ${K^{\circ}}$-model $({\mathscr{X}},{\mathscr{L}})$ of $(X,L)$ consists of an algebraic ${K^{\circ}}$-model ${\mathscr{X}}$ of $X$ and of a line bundle ${\mathscr{L}}$ on ${\mathscr{X}}$ with a fixed isomorphism from ${\mathscr{L}}|_{X}$ to $L$ which we use again for identification. It follows from Vojta’s version of Nagata’s embedding theorem [Voj07, Theorem 5.7] and noetherian approximation that $(X,L)$ has always an algebraic ${K^{\circ}}$-model.