Algebra & Number Theory

Cox rings and pseudoeffective cones of projectivized toric vector bundles

José González, Milena Hering, Sam Payne, and Hendrik Süß

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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.

Article information

Algebra Number Theory, Volume 6, Number 5 (2012), 995-1017.

Received: 7 October 2010
Revised: 20 September 2011
Accepted: 21 December 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14M25: Toric varieties, Newton polyhedra [See also 52B20] 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]

Cox ring pseudoeffective cone toric vector bundle Mori dream space torus quotient Losev–Manin moduli space Deligne–Mumford moduli space iterated blow up


González, José; Hering, Milena; Payne, Sam; Süß, Hendrik. Cox rings and pseudoeffective cones of projectivized toric vector bundles. Algebra Number Theory 6 (2012), no. 5, 995--1017. doi:10.2140/ant.2012.6.995.

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