The diminished base locus is not always closed
2014 ◽
Vol 150
(10)
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pp. 1729-1741
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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.
1963 ◽
Vol 59
(3)
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pp. 555-558
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2000 ◽
Vol 41
(1)
◽
pp. 133-136
1988 ◽
Vol 62
(03)
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pp. 419-423
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