Suppose that $\mathscr{X}$ and $\mathscr{L}$ are defined over a subring $R$ of ${K^{\circ}}$ by a line bundle $\mathscr{L}_{R}$ on a projective regular integral scheme $\mathscr{X}_{R}$ over $R$. We assume furthermore that $R$ is a discrete valuation ring which is defined geometrically by a $d$-dimensional normal variety $B$ over a field $k$, i.e. there exist $b\in B^{(1)}$ and an isomorphism $h\colon R\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{O}_{B,b}$. We read the isomorphism $h$ as an identification. Then Assumption 7.1 is equivalent to the existence of data $(R,k,B,b,h,\mathscr{X}_{R},\mathscr{L}_{R})$ as above assuming furthermore that the field $k$ is perfect and the restriction of $\mathscr{L}_{R}$ to the generic fiber $\mathscr{X}_{R,\eta}$ over $R$ extends to an ample line bundle $\mathcal{A}_{R}$ on $\mathscr{X}_{R}$.

One direction of the equivalence is clear by base change from $B$ to $\Spec\mathcal{O}_{B,b}$. On the other hand, replacing $B$ by an open affine neighbourhood of $b$, it is clear by [EGAIV, Cor. 9.6.4] that $\mathcal{A}_{R}$ extends to an ample line bundle $\mathcal{A}_{B}$ on a projective integral scheme $\mathscr{X}_{B}$ over $B$ and that $\mathscr{L}_{R}$ extends to a line bundle $\mathscr{L}_{B}$ on $\mathscr{X}_{B}$. Since the regular locus of $\mathscr{X}_{B}$ is open [GW10, Cor. 12.52] and since the fiber of $\mathscr{X}_{B}$ over $b$ is contained in the regular locus, we may assume that $\mathscr{X}_{B}$ is also regular by shrinking $B$ again.