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arXiv · 1212.3738

The diminished base locus is not always closed

John Lesieutre
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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.

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John Lesieutre. 2014-04-05. The diminished base locus is not always closed. https://doi.org/10.1112/s0010437x14007544

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