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Let $X$ a smooth projective $3$-fold. Assume that $X$ admits a finite rational map $f:X\dashrightarrow Y$ where $Y$ is a smooth Calabi-Yau 3-fold, and a fibration $g:X\rightarrow \mathbb{P}^2$ with a smooth curve of positive Kodaira dimension as general fiber.

Under these hypothesis what can we say on the Kodaira dimension of $X$ ?

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    $\begingroup$ Not much. The second condition is essentially trivial (take any sufficiently ample line bundle $L$ on $X$, and any general plane in $\mid L\mid$). The first one just tells you that the Kodaira dimension is nonnegative. $\endgroup$
    – abx
    Commented Jul 3, 2017 at 4:08

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