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Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre?

Can this be done with $C\cong \mathbb P^1$?

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  • $\begingroup$ At least in characteristic not 2 or 3 this is impossible: the elliptic fibration is unique for Kodaira dimension 1. For char 2 or 3 you may have to consider quasi-elliptic fibrations, and I haven't thought through it. $\endgroup$
    – Abhinav Kumar
    May 9, 2014 at 3:34
  • $\begingroup$ That answers my question (in which I should have emphasized that I work in characteristic zero). Thank you very much. $\endgroup$
    – Ste3an
    May 9, 2014 at 6:35

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