Let $X$ be a smooth and projective variety over a finite field (separated, finite type, integral). Then after performing a number of blowups I should be able to find a proper surjective map from $X$ to a smooth curve, where the generic fibre is smooth and geometrically integral.
My question is really: how? I assume we can use projection away from a hyperplane, yielding a rational map from $X$ to the projective line (if I'm not making any dimension mistakes here). After performing a number of blowups this becomes an everywhere-defined map (right?). But how can I make sure the generic fibre is smooth and geometrically integral?
As you can see I think I have an intuition for how things should work. But I'd like to grasp it in all mathematical detail, not just a geometric image in my head. If you would have a general reference for such facts, that would be great already.