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Effective Cones of Cycles on Blow-ups of Projective Space

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journal contribution
posted on 19.01.2017 by Izzet Coskun, John Lesieutre, John Christian Ottem
In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up Xn r of P n, n > 4, at r very general points, the cone of divisors is not finitely generated as soon as r > n + 3, whereas the cone of curves is generated by the classes of lines if r ≤ 2 n. In fact, if Xn r is a Mori Dream Space then all the effective cones of cycles on Xn r are finitely generated.


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This is a copy of an article published in the Algebra and Number Theory © 2016 Mathematical Sciences Publishers. DOI:10.2140/ant.2016.10.1983


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