The diminished base locus is not always closed

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Abstract

We exhibit a pseudoeffective ℝ-divisor Dλ 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- (Dλ) = ∪A ample B(Dλ + A) is not closed and that Dλ 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.

Original languageEnglish (US)
Pages (from-to)1729-1741
Number of pages13
JournalCompositio Mathematica
Volume150
Issue number10
DOIs
StatePublished - Oct 2 2014

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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