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The diminished base locus is not always closed

Part of: Birational geometry

Published online by Cambridge University Press:  15 September 2014

John Lesieutre
John Lesieutre*
Affiliation:
Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email johnl@math.mit.edu
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Abstract

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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.

Keywords

diminished base locusZariski decompositionCremona transformations

MSC classification

Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 10 , October 2014 , pp. 1729 - 1741
Copyright
© The Author 2014 

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