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Let $X_{k}$ be a stable curve over a algebraically closed field $k$. We can find a complete DVR $R$ and deform $X_{k}$. Then we can obtain a stable curve $X$ over $R$ whose generic fiber $X_{\overline \eta}$ is a smooth curve and the special fiber is isomorphic to $X_{k}$.

My question is that how to do conversely. This mean is that for example, if given a smooth curve $X_{Q}$ with $g(X_{Q}) \geq 2$, where $Q$ is rational number field, does there exist a prime number $p$ such that $X_{F_{p}}$ is a reduction that we need?

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Could you be a bit more precise? Is $K$ the fraction field of a complete DVR $R$? If so, then the semistable model over (some finite flat extension of) $\text{Spec}(R)$ is uniquely determined by $X_K$ -- you don't get to "construct a reduction". –  Jason Starr Jul 13 '13 at 8:02
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