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 Xrn of 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 2n. In fact, if Xrn is a Mori dream space then all the effective cones of cycles on Xrn are finitely generated.

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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