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Are there complex projective elliptic threefolds $f: X \to S$ with infinitely many rational sections satisfying the following properties?

  • $S$ is a smooth surface of Picard rank $1$.
  • $X$ is smooth with $q(X) = 0$. (Bonus: $\kappa(X) \ge 0$.)
  • The discriminant locus of $f$ in $S$ is smooth.

If we drop any of the conditions above, there are examples of such elliptic threefolds.

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    $\begingroup$ The last condition is pretty strong. It implies the singular fibers are equisingular I suppose. Do you know any example such that the last condition holds and does not come from taking fiber product of an elliptic surface with a trivial family? $\endgroup$
    – AG learner
    Commented May 21, 2023 at 13:28

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