#### Vol. 10, No. 9, 2016

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Effective cones of cycles on blowups of projective space

### Izzet Coskun, John Lesieutre and John Christian Ottem

Vol. 10 (2016), No. 9, 1983–2014
DOI: 10.2140/ant.2016.10.1983
##### Abstract

In this paper we study the cones of higher codimension (pseudo)effective cycles on point blowups 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 blowup ${X}_{r}^{n}$ 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 $r\le {2}^{n}$. In fact, if ${X}_{r}^{n}$ is a Mori dream space then all the effective cones of cycles on ${X}_{r}^{n}$ are finitely generated.

##### Keywords
Cones of effective cycles, higher codimension cycles, blowups of projective space, Mori dream space
##### Mathematical Subject Classification 2010
Primary: 14C25, 14C99
Secondary: 14E07, 14E30, 14M07, 14N99
##### Milestones
Revised: 11 July 2016
Accepted: 21 August 2016
Published: 22 November 2016
##### Authors
 Izzet Coskun Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607 United States John Lesieutre Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607 United States John Christian Ottem Department of Mathematics University of Oslo Blindern 0316 Oslo Norway