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Let $C$ be a semi-stable curve over $Spec(O_K)$ where $K$ is an algebraic number field.

Assume that $X_s$ is singular for $s\in Spec(O_K)$. How do we know that there exists a finite separable extension $k'$ of $k(s)$ such that the singular points of $X_s\times_{Spec(k(s))}Spec(k')$ are split?

On Liu's book "Algebraic Geometry and Arithmetic Curves", p514, Cor 3.22 (a)'s proof, it said that 'by the virtue of Prop 3.7(c)' which makes me confused.

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