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This question is motivated by complex algebraic geometry.

If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the linear system $|mK_X|$ for $m$ divisible and large enough) has its general fibre of Kodaira dimension $0$.

This is not the case in char. $p$. In char. $2$ and $3$, there are quasi-elliptic fibrations where the general fibres are rational curves with a single cusp.

Question: Are these (elliptic curve & rational curve with cusp) the only possibilities of general fibres in a codimension-1 Iitaka fibration?

More generally, what are the possibilities of the general fibres in an Iitaka fibration?

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  • $\begingroup$ I think those are the only two possibilities. $\endgroup$
    – Jason Starr
    Commented May 27, 2023 at 16:57

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