Does anyone know examples of a numerically non trivial nef divisor with Iitaka dimension 0? (Unfortunately, this question might be trivial as right now I can't think of any effective divisors with Iitaka dim 0 other than exceptional divisors)



Example of nef divisor with Iitaka dimension 0
Here's an example that works over an uncountable field. 


Let $E$ be an elliptic curve in $\mathbf P^2$ and let $p\_1, \ldots, p\_9$ be 9 points on $E$. Blowing up these points, the proper transform $C$ of $E$ is an irreducible curve with self-intersection 0, hence nef.


Notice that if some multiple $mC$ of $C$ moves, it must move to a curve disjoint from $C$. But that implies that the corresponding power $N\_C^m$ of the normal bundle of $C$ is trivial.


Now by choosing the points $p\_1, \ldots, p\_9$ on $E$ appropriately, we can arrange that $N\_C$ is any desired element of $\operatorname{Pic}^0(C)$. In particular since our field is uncountable we can choose it to be a non-torsion element, so no power $N\_C^m$ is trivial, and therefore no multiple $mC$ moves. 


