Let $K$ be a number field and $\mathscr{X} \longrightarrow \mathcal{O}_K$ be an arithmetic surface (smooth and projective). Let $D$ be a horizontal divisor.
We know $D=\overline{\{x\}}$ for some closed point $x$ in the generic fiber $\mathscr{X}_K$. We know $D=\operatorname{Spec}(R)$ where $R$ is an order in the number field $L=K(D)=\kappa(x)$.
From $x: \operatorname{Spec}(L) \longrightarrow \mathscr{X}_K$, valuative criterion gives us $\varepsilon: \operatorname{Spec}(\mathcal{O}_L) \longrightarrow \mathscr{X}$ whose image is $D=\operatorname{Spec}(R)$. This map is in general only finite and not being a closed immersion.
Under this setting, $D$ is regular $\Longleftrightarrow$ the order $R=\mathcal{O}_L$ $\Longleftrightarrow$ the morphism $\varepsilon: \operatorname{Spec}(\mathcal{O}_L) \longrightarrow \mathscr{X}$ is a closed immersion.
My question is:
Since horizontal divisors and closed points of generic fibres are in $1:1$ correspondence, I want a criterion, in terms of the point $x: \operatorname{Spec}(L) \longrightarrow \mathscr{X}_K$, such that $\varepsilon: \operatorname{Spec}(\mathcal{O}_L) \longrightarrow \mathscr{X}$ is a closed immersion.