Cox ring

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In algebraic geometry, a Cox ring is a sort of universal homogeneous coordinate ring for a projective variety, and is (roughly speaking) a direct sum of the spaces of sections of all isomorphism classes of line bundles. Cox rings were introduced by Hu & Keel (2000), based on an earlier construction by David A. Cox in 1995 for toric varieties.


  • Cox, David A. (1995), "The homogeneous coordinate ring of a toric variety", J. Algebraic Geom., 4 (1): 17–50, MR 1299003
  • Hu, Yi; Keel, Sean (2000), "Mori dream spaces and GIT", Michigan Math. J., 48: 331–348, arXiv:math/0004017, doi:10.1307/mmj/1030132722, MR 1786494
  • Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, Jürgen; Laface, Antonio (2015), Cox Rings, Cambridge Studies in Advanced Mathematics, vol. 144 (1st ed.), Cambridge: Cambridge University Press, ISBN 978-1-107-02462-5, MR 3307753