Does there exist a smooth projective variety $X$ such that $\operatorname{NEF}(X)-\{0\}$ is strictly contained in $\operatorname{Big}(X)$, where $\operatorname{Big}(X)$ is the interior of $\overline{\operatorname{Eff}(X)}$? 



$\operatorname{NEF}(X)\subset\operatorname{Big}(X)$?
Probably someone will come up with a simpler example, but here's a Calabi-Yau threefold that fits the bill: look at <https://arxiv.org/pdf/1206.1649.pdf> . Use the example given in section 6. The Picard rank is 2, the nef cone is given in Prop 6.1, and the rays on the boundary of the movable cone are computed in Lemma 6.7. In this case the nef cone is strictly contained in the movable cone, which is a subcone of the pseudoeffective cone in general, although in this example they should agree.


