Cox ring

{{Short description|Universal homogenous coordinate ring of a projective variety}}

In algebraic geometry, a Cox ring (or total coordinate 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 {{harvtxt|Hu|Keel|2000}}, based on an earlier construction by David A. Cox in 1995 for toric varieties.

References

{{citation|mr=1299003|last=Cox|first=David A.|authorlink=David A. Cox|title=The homogeneous coordinate ring of a toric variety|journal=J. Algebraic Geom.|volume= 4 |year=1995|issue= 1|pages= 17–50}}
{{citation|mr=1786494|last1=Hu|first1= Yi|last2= Keel|first2= Sean|title=Mori dream spaces and GIT

|journal=Michigan Math. J.|volume= 48 |year=2000|pages=331–348|doi=10.1307/mmj/1030132722|arxiv=math/0004017}}

{{Citation | last1=Arzhantsev | first1=Ivan | last2=Derenthal | first2=Ulrich | last3=Hausen | first3=Jürgen | last4=Laface | first4=Antonio | title=Cox Rings | publisher=Cambridge University Press | location=Cambridge | edition=1st | series=Cambridge Studies in Advanced Mathematics | isbn=978-1-107-02462-5 | mr=3307753 | year=2015 | volume=144}}