Effective cones of cycles on blowups of projective space

Coskun, Izzet; Lesieutre, John; Ottem, John

doi:10.2140/ant.2016.10.1983

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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 $r \leq 2^{n}$ . 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

Received: 15 March 2016

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

Vol. 10, No. 9, 2016