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 $f$?



Sections of proper, flat morphism
I am certain that I have answered this before. Let $Y$ be $\mathbb{P}^2$. Let $\Omega\_{\mathbb{P}^2/k}$ be the sheaf of relative differentials. This is locally free of rank $2$. Via the Euler sequence and the Whitney sum formula, the total Chern class of $\Omega\_{\mathbb{P}^2/k}$ equals
$$
c\_t(\Omega\_{\mathbb{P}^2/k}) = 1 - 3c\_1(\mathcal{O}(1))t + 3c\_1(\mathcal{O}(1))^2t^2.
$$
If the K-theory class $[\Omega\_{\mathbb{P}^2/k}]$ of this rank two sheaf were a sum of the classes of two invertible sheaves, then those invertible sheaves would be $[\mathcal{O}(a)]$ and $[\mathcal{O}(b)]$ for $a,b\in \mathbb{Z}$ (by the classification of invertible sheaves on projective spaces). Again by the Whitney sum formula, this would give an identity in the Chow groups,
$$
1 - 3c\_1(\mathcal{O}(1))t + 3c\_1(\mathcal{O}(1))^2 t^2 = \left(1+ac\_1(\mathcal{O}(1))t\right)\cdot \left(1+bc\_1(\mathcal{O}(1))t\right).
$$
Since $1$, $c\_1(\mathcal{O}(1))$ and $c\_1(\mathcal{O}(1))^2$ are an additive basis for the Chow group of $\mathbb{P}^2$, this gives identities of integers,
$$
a+b = -3, \ \ ab = 3.
$$
This has no solutions. Thus, $\Omega\_{\mathbb{P}^2/k}$ fits into no short exact sequence
$$
0 \to \mathcal{O}(a) \to \Omega\_{\mathbb{P}^2/k} \to \mathcal{O}(b) \to 0.
$$
Now let $f:X\to Y$ be the $\mathbb{P}^1$-bundle associated to this locally free sheaf,
$$
X = \text{Proj}\_Y(\text{Sym}^{\bullet}\_{\mathcal{O}\_Y}(\Omega\_{Y/k})).
$$
If there were a global section $s$ of $f$, then the pullback of the universal quotient invertible sheaf of $f^\*\Omega\_{Y/k}$ would give a short exact sequence as above. Thus, there is no global section of $f$. However, there are plenty of rational sections.


**Edit.** I looked through my answers. The construction above is also in my answer to the following: [What can we say about this generalization of simply-connectedness?](https://mathoverflow.net/questions/124386/what-can-we-say-about-this-generalization-of-simply-connectedness/124400#124400).


