The diminished base locus is not always closed

We exhibit a pseudoeffective ℝ-divisor D_λ on the blow-up of double-struck P³ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus B- (D_λ) = ∪_{A ample} B(D_λ + A) is not closed and that D_λ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an ℝ-divisor on the family of blow-ups of double-struck P² at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.