Sections of proper, flat morphism

Question

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 $f$?

Answer 1

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?.