I am aware that if an elliptic surface contains multiple fibers, then it has no section. Is the converse false? 


In particular, I am looking for an example of a projective, properly elliptic surface (Kodaira dimension 1), fibered over $\mathbb{P}^1$, with no multiple fibers and no section.



Properly elliptic surface with no multiple fibers and without a section
Please confer Corollary 2.2 of the following with $d$ equal to $3$ and with $n$ equal to $2$.


Jason Starr   

A pencil of Enriques surfaces of index 1 with no section   

<https://arxiv.org/pdf/math/0602639.pdf>


This proves that for every integer $e\geq 2$, for a very general hypersurface $X$ in $\mathbb{P}^2\times \mathbb{P}^1$ in the complete linear system of $\mathcal{O}\_{\mathbb{P}^2\times \mathbb{P}^1}(3,e)$, there is no rational section of the projection, $$\text{pr}\_2|\_X:X\to \mathbb{P}^1.$$ Since the locus of multiple curves in the complete linear system $\mathcal{O}\_{\mathbb{P}^2}(3)$ has codimension $3$, for a general $X$ in the complete linear system, there are no multiple fibers.


By adjunction, the dualizing sheaf of $X$ equals $\mathcal{O}\_{\mathbb{P}^2\times \mathbb{P}^1}(0,e-2)|\_X$. Thus, for $e\geq 3$, the dualizing sheaf is the pullback of an ample sheaf by $\text{pr}\_2|\_X$. In that case, the Kodaira dimension equals $1$.


