Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is of codimension $2$ such that the total space $f^{-1}(U)$ is non-singular and such that $f:f^{-1}(U) \to U$ has geometrically irreducible fibers. Denote by $\eta$ the generic point of $Y$. Suppose that there exists a morphism from $\eta$ to the generic fiber $X_{\eta}$. Then, does there exist a morphism $s:Y \to X$ such that $f \circ s$ is identity on $Y$?

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is of codimension $2$ such that the total space $f^{-1}(U)$ is non-singular and such that $f:f^{-1}(U) \to U$ has geometrically irreducible fibers. Denote by $\eta$ the generic point of $Y$. Suppose that there exists a morphism from $\eta$ to the generic fiber $X_{\eta}$. Then, does there exist a morphism $s:Y \to X$ such that $f \circ s$ is identity on $Y$?

EDIT: Assumptions as above. Then, does there exist a rational section of $f$?