The total coordinate ring of a normal projective variety
Abstract
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.
Keywords
Cite
@article{arxiv.math/0305354,
title = {The total coordinate ring of a normal projective variety},
author = {E. Javier Elizondo and Kazuhiko Kurano and Kei-ichi Watanabe},
journal= {arXiv preprint arXiv:math/0305354},
year = {2007}
}
Comments
This is the final version that will appear in the Journal of Algebra. 11 pages. LaTex