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Feb 18, 2018 at 0:49 comment added Chen Jiang $K_B=f^*K_A+E$, where $E$ is an effective $f$-exceptional divisor supported on $D$. Hence $K_B+D$ has same Kodaira dimension as $K_A$ which is $0$. But of course it is not trivial.
Feb 17, 2018 at 21:05 history asked Gonal_curve CC BY-SA 3.0