  We show that the augmented base locus coincides with the exceptional locus
(i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme
projective over a field (of any characteristic). Next we prove a semi-ampleness
criterion in terms of the augmented base locus generalizing a result of Keel.
We also study nef divisors with positive top intersection number, and discuss
some problems related to augmented base loci of log divisors.
