The diminished base locus is not always closed

TL;DR

This paper provides counterexamples showing that the diminished base locus of certain divisors on blow-ups of projective spaces can be non-closed, challenging previous assumptions about their geometric properties.

Contribution

It constructs explicit divisors on blow-ups of P^3 and P^2 that demonstrate non-closed diminished base loci and failure of Zariski decompositions, revealing new complexities in the geometry of divisors.

Findings

D__ on blow-up of P^3 has non-closed diminished base locus

Constructed divisor on blow-up of P^2 is nef on general fibers but not over special divisors

Counterexamples to assumptions about Zariski decompositions and base locus closure

Abstract

We exhibit a pseudoeffective R-divisor D_\lambda on the blow-up of 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 B_-(D_\lambda) = \bigcup_{A ample}} 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 R-divisor on the family of blow-ups of 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.