On the Numerical Dimension of Pseudo-Effective Divisors in Positive Characteristic

Let $X$ be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if $D$ is a pseudo-effective ${\bf R}$-divisor on $X$ which is not numerically equivalent to the negative part in its divisorial Zariski decomposition, then the numerical dimension of $D$ is positive. In characteristic zero, this was proved by Nakayama using vanishing theorems.