Let $k$ be a field and $X$ be an irreducible algebraic variety over $k$. Let $Y$ be a variety over $k(X)$. Is it true that we have an algebraic variety $Z$ over $k$ and a morphism $f:Z \to X$ over $k$ whose generic fiber is $Y$?
For instance $y^2 = x(x-1)(x-t)$ is an elliptic curve over $k(t)$, but I think that it also gives a surface $Z$ over $k$ and $f : (x,y,t) \mapsto t \in X = \Bbb P^1_k$. I'm not sure how to proceed in the general case.