Let $\overline{NE}(X)$ be the closure of the cone generated by the numerical classes of effective curves and $\overline{\mathrm{Mov}}(X)$ the closure of the cone of moving curves.

(Q1) Is there an example of a smooth projective variety $X$ such that

• $\overline{NE}(X)$ is (finite) polyhedral, but
• $\overline{\mathrm{Mov}}(X)$ is not?

Here are some trivial observation:

1

If $X$ is a surface then the two cones are dual to each other and hence they are either both polyhedral or not.

2

If $X$ is a Fano variety then

• $\overline{NE}(X)$ is polyhedral by the Cone Theorem, and
• $\overline{\mathrm{Mov}}(X)$ is polyhedral if $\dim \leq4$ by a result of Barkowski (see here).

So, perhaps I should really ask:

(Q2) Is it true that $\overline{NE}(X)$ is polyhedral if and only if $\overline{\mathrm{Mov}}(X)$ is polyhedral?

If true, this would (for example) provide a proof of Barkowski's result in all dimensions.