The definition I most often see for what it means for a projective variety $X$ over a field $k$ to be rationally connected is that there exists a variety $M$ and a dominant morphism $f:\mathbb{P}^1\times\mathbb{P}^1\times M \to X\times X$. For instance, this is the definition given in both Kollar's "Rational curves on algebraic varieties" and Debarre's "Higher-dimensional algebraic geometry". Both books claim that, for $k$ algebraically closed, this implies the existence of a rational curve connecting any two general geometric points. However, I don't understand why the field needs to be algebraically closed for this to be true, it seems that a pretty straightforward argument could be made for arbitrary base field.

With this in mind, could anyone provide any examples of rationally connected varieties which don't necessarily possess rational curves between a general pair of points?

closed(rather than algebraically complete). $\endgroup$ – Mikhail Borovoi Apr 25 '16 at 22:03