Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.


Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)


Is $X(S)$ non-empty?


The answer is positive if $S$ is of dimension at most one.



Are generically trivial finite unramified morphisms trivial
The answer is yes in far more generality; see Prop. 6.2 of Liu-Lorenzini-Gabber <http://arxiv.org/abs/1404.5366>


In your case you can take the closure of a generic section and use Zariski's main theorem as Jason Starr alluded to above.


