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Let $f:X\rightarrow S$ be a morphism of varieties (say, over an algebraically closed field $k$). Is the locus of points above which the fibers of $f$ are rationally connected $\{s\in S:X_s \text{ is rationally connected}\}$ a constructible set in $S$? I'm wondering about this question because I read in an article a proof where the author seemed to have used the fact the the fiber of a morphism $g:V\rightarrow \mathbb{A}_k^n$ above every dimension $1$ point is rationally connected if we know that the generic fiber is rationally connected. I can see it can be deduced if being rational connected is a constructible property. If not, how do we transfer the rational connectedness to general fibers from the generic fiber as above?

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  • $\begingroup$ Typically in order to talk about rationally connected varieties, one works with varieties which are smooth and projective (so $g:V\to\Bbb A^n$ is a little surprising), where the set of points over where the fiber is rationally connected is clopen. Generally this result is one that's proven very early on - Kollar's book should have this fairly early, or these lecture notes by Harris have the proof inside the first 3 pages, for instance. $\endgroup$ – KReiser Oct 19 '18 at 18:17

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