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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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I am having trouble parsing this question. Could you phrase this a little differently, or perhaps give an example? – Jason Starr Apr 6 '13 at 1:00
@Starr: I have modified the question. I hope it makes more sense now. – Jana Apr 6 '13 at 1:24
I already deleted a comment where I misunderstood the question. It's hard to understand what you are really asking. Are you asking whether given $b,p$ as above, there exists a section of the family passing through $p$ and through a rational point in the fiber above $c$? I think the answer is no. – Felipe Voloch Apr 6 '13 at 1:59
@Voloch: I am asking something similar. I am roughly asking that if a variety with a rational point deforms, when does the rational point deform with it? I think it is equivalent to what you say. I know that this is not true in general. So I am asking for conditions to impose on the members of the family so as to get a positive answer. – Jana Apr 6 '13 at 2:28
@Jana: Even though rationally connected varieties deform to rationally connected varieties, it is not true that rational curves in rationally connected varieties always deform in families. – ulrich Apr 6 '13 at 4:42

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