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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.

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    $\begingroup$ Do you require the surface to be projective? (Maybe that is part of "properly".) If Kaehler, non-projective surfaces are acceptable, then you can take the base change of a non-projective elliptic K3 by a high degree morphism from $\mathbb{P}^1$ to $\mathbb{P}^1$. $\endgroup$
    – Jason Starr
    Jul 11, 2018 at 17:10
  • $\begingroup$ I do need it to be projective. Edited accordingly. $\endgroup$
    – user564401
    Jul 11, 2018 at 21:19
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    $\begingroup$ Possibly you can use the infinitesimal approach to Noether-Lefschetz theory to prove that a very general hypersurface in $\mathbb{P}^2\times \mathbb{P}^2$ of bidegree $(3,d)$, $d\gg 0$, has no section (obviously it has no multiple fibers, since that is codimension $>2$ in the parameter space of plane cubic curves). $\endgroup$
    – Jason Starr
    Jul 12, 2018 at 0:14
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    $\begingroup$ @JasonStarr Can you point me toward a reference? Also, it's not clear to me how this would get one closer to an elliptic surface, but maybe I'm just not seeing it. The only way I can think of is to start with an elliptic surface without a section and with multiple fibers, then do a log transformation. But then of course, determining what one ends up with is nontrivial... Curious to see if anyone can point me toward a specific example of this. $\endgroup$
    – user564401
    Jul 12, 2018 at 16:16
  • $\begingroup$ I meant to write $\mathbb{P}^2\times \mathbb{P}^1$, not $\mathbb{P}^2\times \mathbb{P}^2$. Sorry about that. $\endgroup$
    – Jason Starr
    Jul 12, 2018 at 16:54

1 Answer 1

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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 $1$.

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