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