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I think following post in the mathoverflow gives an answer:

Effective versus movable cones of curvesEffective versus movable cones of curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.

I think following post in the mathoverflow gives an answer:

Effective versus movable cones of curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.

I think following post in the mathoverflow gives an answer:

Effective versus movable cones of curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.

Source Link

I think following post in the mathoverflow gives an answer:

Effective versus movable cones of curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.