The total coordinate ring of a normal projective variety

Details The total coordinate ring of a normal projective variety The total coordinate ring of a normal projective variety

2003-05-25

arxiv.org/abs/math/0305354v2

J. Algebra. Vol. 276, Issue 2 , June 2004, Pages 625-637

Mathematics

Algebraic Geometry

This is the final version that will appear in the Journal of Algebra. 11 pages. LaTex

Scientific paper

The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.

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