# Is rational connectedness a constructible property?

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?

• 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. – KReiser Oct 19 '18 at 18:17