Homogeneous coordinates for algebraic varieties

By F. Berchtold, Jürgen Hausen. 2002

56 citations recorded by the source corpus.

Abstract We associate to every divisorial (e.g., smooth) variety X with only constant invertible global functions and finitely generated Picard group a Pic(X)-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties X with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3de8cbcdce73f9d43aff25396d2c62d83503fa91

How researchers see this paper

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In Algebra and Number Theory

1. P. Hacking
2. S. Keel
3. Jürgen Hausen
4. Mark Gross
5. Hendrik Suss
6. Antonio Laface
7. M. Artebani
8. F. Berchtold

In Geometry and Topology

1. Jürgen Hausen
2. P. Hacking
3. S. Keel
4. Mark Gross
5. Hendrik Suss
6. F. Berchtold
7. Antonio Laface
8. M. Artebani

In Statistical and Nonlinear Physics

1. Jürgen Hausen
2. F. Berchtold
3. P. Hacking
4. Mark Gross
5. S. Keel
6. Hendrik Suss
7. Antonio Laface
8. Ben Davison

Citations in the corpus

• Singular del Pezzo surfaces whose universal torsors are hypersurfaces (2006, unknown): “[HK00], [Muk01], [BH03a], [BH03b], [EKW04], [CT05].”
• 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].”
• On Cox rings of K3 surfaces (2009, background): “If the divisor class group Cl(X) is free, then the Cox ring R(X) admits unique factorization, see [6].”
• Cox Rings and Combinatorics II (2008, background): “gether to the total coordinate ring: R(X) := M D∈K Γ(X,O(D)). For a toric variety X, the total coordinate ring is a polynomial ring [8]. In general, it turns out to be after all a factorial ring, see [3] and, more generally, [9]. The present paper is devoted to the study of varieties X with a finitel”
• The Cox ring of an algebraic variety with torus action (2009, background): “Thus, we infer from [5, Prop.”
• The total coordinate ring of a normal projective variety (2003, background): “s a unique factorization domain for a connected normal Noetherian scheme whose divisor class group is a finitely generated free abelian group. We should mention that independently Berchtold and Hausen [BH02] proved that the total coordinate ring is a unique factorization domain for a locally factoria”
• Birational geometry of cluster algebras (2013, methodology): “is totally coprime, then the upper cluster algebra Γ(A,OA) is factorial. 38 MARKGROSS,PAULHACKING,ANDSEANKEEL Proof. For the cases other than Aprin, this follows from Theorem 1.1 of [Ar08], (see also [BH03], Prop. 8.4.) For the principal case, we note that the map ˜p: Aprin → X is a TN◦-torsor, and ”