Let $X\rightarrow Spec(O_K)$ be a semi-stable curve, where $K$ is an algebraic number field. Let $D$ be a horizontal divisor and finite extension $K'\supset K$ contains the splitting field of the function field of $D$.

Then how do we know that $D':=D\times_{Spec(O_K)}Spec(O_{K'})$ is a section of $X'=X\times_{Spec(O_K)}Spec(O_{K'})$?


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