The Cox ring of an algebraic variety with torus action

Juergen Hausen*, Hendrik Suess

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth K*-surfaces, our results give a description of the Cox ring in terms of Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all Gorenstein del Pezzo K*-surfaces with Picard number at most two and the Cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of Klyachko's description. (C) 2010 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)977-1012
Number of pages36
JournalAdvances in Mathematics
Issue number2
Publication statusPublished - 1 Oct 2010


  • Cox ring
  • Torus action

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