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Sep 6, 2022 at 11:37 comment added Tom Ducat I would guess so. I am not sure how to go about writing down an explicit example, but I could imagine a terminal threefold $X$ with two curves $C_-,C_+\subset X$ with $\pm KC_\pm > 0$ and a birational map $X\dashrightarrow X$ which flips $C_-$ to $C_+$ and antiflips $C_+$ to $C_-$.
Sep 5, 2022 at 16:23 comment added Jérémy Blanc Nice counterexample! Thanks. Do you think we can do this in dimension 3 with $F$ also being some curves?
Sep 5, 2022 at 9:15 history answered Tom Ducat CC BY-SA 4.0