The diminished base locus is not always closed

Abstract

We exhibit a pseudoeffective R-divisor D_λon 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_λ) = \bigcup_{A ample}} B(D_λ+A) is not closed and that D_λdoes 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.

Revamped introduction, various simplifications. To appear in Compositio Mathematica (with minor changes)

References in corpus (2)

• On positivity and semistability of vector bundles in finite and mixed characteristics
• On a class of special linear systems of P^3

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