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 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. 
Paolo Cascini, Christopher Hacon, Mircea Mustata, and Karl Schwede
math.AG