The diminished base locus is not always closed

Abstract

AbstractWe exhibit a pseudoeffective$\mathbb{R}$-divisor${D}_{\lambda }$on the blow-up of${\mathbb{P}}^{3}$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${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$is not closed and that${D}_{\lambda }$does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an$\mathbb{R}$-divisor on the family of blow-ups of${\mathbb{P}}^{2}$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.