#### Vol. 6, No. 5, 2012

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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
##### Abstract

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 $\phantom{\rule{0.3em}{0ex}}{\overline{\phantom{\rule{0.3em}{0ex}}M}}_{0,n}$.

##### Keywords
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