Let $D$ be a nef divisor (moreover suppose it is effective if you prefer) on a normal projective variety of dimension $n$. Let $k\in[1,n1]$. If $D^k\cdot V=0$ for generic subvarieties $V\subseteq X$ of dimension $k$, can I conclude that $D^k\cdot V=0$ for all subvarieties $V$ of dimension $k$? In other words can I conclude that the numerical dimension of $D$ is less than $k$?
