Vol. 6, No. 5, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Cox rings and pseudoeffective cones of projectivized toric vector bundles

José González, Milena Hering, Sam Payne and Hendrik Süß

Vol. 6 (2012), No. 5, 995–1017

We study projectivizations of a special class of toric vector bundles that includes cotangent bundles whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M¯0,n.

Cox ring, pseudoeffective cone, toric vector bundle, Mori dream space, torus quotient, Losev–Manin moduli space, Deligne–Mumford moduli space, iterated blow up
Mathematical Subject Classification 2010
Primary: 14C20
Secondary: 14J60, 14M25, 14L30
Received: 7 October 2010
Revised: 20 September 2011
Accepted: 21 December 2011
Published: 31 July 2012
José González
Department of Mathematics
The University of British Columbia
Room 125, 1984 Mathematics Road
Vancouver BC V6T 1Z2
Milena Hering
Department of Mathematics
University of Connecticut
196 Auditorium Road, Unit 3009
Storrs, CT 06269-3009
United States
Sam Payne
Mathematics Department
Yale University
10 Hillhouse Ave.
New Haven, CT 06511
United States
Hendrik Süß
Lehrstuhl für Algebra und Geometrie
BTU Cottbus
Postfach 10 13 44
D-03013 Cottbus