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The diminished base locus is not always closed
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 10 / October 2014
- Published online by Cambridge University Press:
- 15 September 2014, pp. 1729-1741
- Print publication:
- October 2014
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- Article
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- You have access
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We 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.
