"Smooth" rationally connectivity

Question

Let's say that a variety (projective, over $\mathbb{C}$) is smooth rationally connected if, for every couple of points in it, we can connect those points with a smooth rational curve. What do we know about this condition? Is it implied by smoothness+rationally connectivity, or by unirationality? Thanks, Julian

Answer 1

For smooth projective varieties of dimension at least $3$ over $\mathbb{C}$, this property is actually equivalent to rational connectivity, which is in turn equivalent to rational chain connectivity. A nice discussion of these topics can be found in Chapter 4 of Debarre's book Higher Dimensional Algebraic Geometry.

For singular varieties this is not true anymore, as shown by the projective cone over an elliptic curve. In fact, it is rationally chain connected (any two points can be joined by a chain of two smooth rational curves through the vertex), but it is not rationally connected (there is no irreducible rational curve connecting two points in different lines of the ruling), so in particular it does not satisfy your property.