TITLE: Are rational varieties simply connected?
QUESTION [27 upvotes]: Is it true that every smooth rational variety X is simply connected? How is the proof?
Would it be still true if X has mild (for example orbifold) singularities?

REPLY [3 votes]: In characteristic zero you do not neen rational. It is enough rationally connected. 
Let $X$ be a smooth, projective, rationally connected variety over a field of characteristic zero. Then

Any finite étale cover of $X$ is trivial;
$X$ is simply connected.

You can find this, for instance, in O. Debarre "Higher-Dimensional Algebraic Geometry", Corollary 4.18.