Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$rational point. Fix a pair $(p,\mathcal{X}_b)$ where $p$ is a $K$rational point in $\mathcal{X}_b$ for some closed point $b \in B$. Under what condition is it true that if $\mathcal{X}_b$ deforms to $\mathcal{X}_c$ then the point $p$ deforms to a $K$rational point in $\mathcal{X}_c$? For example as far as I understand a rationally connected variety deforms to a rationally connected variety. I am asking a similar question for rational points.
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