The total coordinate ring of a normal projective variety

By J. Elizondo, Kazuhiko Kurano, Kei-ichi Watanabe. 2003

69 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ea3f8155aad639fa04bfb5e641133e91ea33426f

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Citations in the corpus

GIT-Equivalence beyond the ample cone (2005, background): “An important property of the total coordinate ring R(X) is that it is a factorial ring; see [1] and [11].”
Hilbert's 14th problem and Cox rings (2005, background): “−1] is the desired inverse. Notation 3.2. In this section, X= Xa,b,c. Proposition 3.3. Cox(X) is a unique factorization domain. Proof. The Cox ring of a normal projective variety is known to be a UFD [EKW]. We can also use a simple observation: the ring of invariants of a UFD with respect to the act”
Cox Rings and Combinatorics II (2008, background): “e proof will be split into two parts. An important ingredience of the ﬁrst part is factoriality of the total coordinate ring. This can be proved using the results of [2]; for a diﬀerent approach, see [7]: Proposition 4.3. Let X be a normal variety with O ∗(X) = K and ﬁnitely generated free divisor c”
Cox rings and combinatorics (2003, background): “A G ] 1 0 D ec Introduction The remarkable success of Cox’s homogeneous coordinate ring in toric geometry [8] has stimulated many authors to study related rings for more general varieties, see [1], [3], [9], [16] and [17].”
Singular del Pezzo surfaces whose universal torsors are hypersurfaces (2006, unknown): “[HK00], [Muk01], [BH03a], [BH03b], [EKW04], [CT05].”
The total coordinate ring of a wonderful variety (2006, background): “Thus, we may define (after [10, 14, 18]) the total coordinate ring R(X) as the direct sum of spaces of sections of all linear combinations of colors with integral coefficients.”