Let $Y$ be a projective variety over $F$ and let $X\coloneqq Y\otimes_{F}K$.

• (a)

  Any $K^{\circ}$-model of $X$ is dominated by the base change of a projective $R$-model of $Y$ to $K^{\circ}$.

• (b)

  If a projective $K^{\circ}$-model ${{\mathscr{X}}}$ of $X$ dominates ${\mathscr{Y}}\otimes_{R}K^{\circ}$ for a projective $R$-model ${\mathscr{Y}}$ of $Y$, then ${{\mathscr{X}}}\simeq{\mathscr{Y}}^{\prime}\otimes_{R}K^{\circ}$ for a projective $R$-model ${\mathscr{Y}}^{\prime}$ of $Y$ dominating ${\mathscr{Y}}$.

• (c)

  Every model function on ${X^{{\mathrm{an}}}}$ is defined over $R$.