Cox Rings

Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, Juergen; Laface, Antonio

doi:10.48550/arxiv.1003.4229

Cited by 23 publications

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“…Here we present the combinatorial approach to varieties with finitely generated Cox ring developed in [10,19], see also [4,Chap. III].…”

Section: Cox Rings and Combinatoricsmentioning

confidence: 99%

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Three lectures on Cox rings

Hausen¹

2013

Torsors, Étale Homotopy and Applications to Rational Points

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IntroductionThese are notes of an introductory course held at the conference "Torsors: Theory and Applications" in Edinburgh, January 2011. Cox rings play meanwhile an important role in Arithmetic and Algebraic Geometry, and in fact appeared independently in these fields [12,13,25]. We present basic ideas and concepts, in particular, we treat the interaction with Geometric Invariant Theory. The notes are kept as a survey; for details and proofs we refer to [4].The first lecture begins with a rigorous definition of the Cox sheaf R of a variety X as a sheaf of algebras graded by the divisor class group Cl(X). The Cox ring is then the algebra R(X) of global sections. After recalling the basic correspondence between graded algebras and quasitorus actions, we discuss the relative spectrum X := Spec X R of the Cox sheaf. We call X → X the characteristic space; it coincides with the universal torsor [12,38] precisely for locally factorial varieties X. The first lecture ends with a characterization of X in terms of Geometric Invariant Theory.The aim of the second lecture is to present a machinery for encoding varieties via their Cox ring. A first input is an explicit description of the variation of not necessarily quasiprojective good quotients for quasitorus actions on affine varieties. This allows to encode torically embeddable varieties with finitely generated Cox ring in terms of what we call "bunched rings". Specialized to the case of toric varieties [15,33,18], the bunched ring data correspond to fans via Gale duality. Moreover, the language of bunched rings extends basic features of techniques developed for weighted complete intersections [17,26] to the multigraded case. We show how to read off basic geometric properties from the defining data and briefly touch the relations to Mori Theory.The third Lecture is devoted to varieties with torus action. Generalizing the case of a toric variety, we describe the Cox ring in terms of the action. In the case of a complexity one action, one obtains a very explicit presentation in terms of trinomial relations. Combining this with the language of bunched rings leads to a concrete description of rational complete varieties with a complexity one action turning them into an easy accessible class for concrete computations. We demonstrate this in the case of K * -surfaces. Some classification results on Fano threefolds and del Pezzo surfaces based on this approach are included in the text.

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“…Here we present the combinatorial approach to varieties with finitely generated Cox ring developed in [10,19], see also [4,Chap. III].…”

Section: Cox Rings and Combinatoricsmentioning

confidence: 99%

“…We present basic ideas and concepts, in particular, we treat the interaction with Geometric Invariant Theory. The notes are kept as a survey; for details and proofs we refer to [4].…”

Section: Introductionmentioning

confidence: 99%

Three lectures on Cox rings

Hausen¹

2013

Torsors, Étale Homotopy and Applications to Rational Points

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“…For free K, the properties factorial and factorially graded are equivalent [3], but for a K with torsion the latter is more general. Our motivation to study factorially graded algebras is that the Cox rings of algebraic varieties are of this type, see for example [2].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…see [2] for the details of the precise definition. As mentioned, our aim is to write down all possible Cox rings of normal rational complete varieties with a complexity one torus action.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Factorially graded rings of complexity one

Hausen

Herppich

2013

Torsors, Étale Homotopy and Applications to Rational Points

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Abstract. We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for homogeneous elements. Our main results describe these algebras in terms of generators and relations. We apply this to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action. Statement of the resultsThe subject of this note are finitely generated normal algebras R = ⊕ K R w over some algebraically closed field K of characteristic zero graded by a finitely generated abelian group K. We are interested in the following homogeneous version of a unique factorization domain: R is called factorially (K-)graded if every homogeneous nonzero nonunit is a product of K-primes, where a K-prime element is a homogeneous nonzero nonunit f ∈ R with the property that whenever f divides a product of homogeneous elements, then it divides one of the factors. For free K, the properties factorial and factorially graded are equivalent [3], but for a K with torsion the latter is more general. Our motivation to study factorially graded algebras is that the Cox rings of algebraic varieties are of this type, see for example [2].We focus on effective K-gradings of complexity one, i.e., the w ∈ K with R w = 0 generate K and K is of rank dim(R) − 1. Moreover, we suppose that the grading is pointed in the sense that R 0 = K holds. The case of a free grading group K and hence factorial R was treated in [5, Section 1]. Here we settle the more general case of factorial gradings allowing torsion. Our results enable us to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action. This complements [6], where the Cox ring of a given variety was computed in terms of the torus action.In order to state our results, let us fix the notation. For r ≥ 1, let A = (a 0 , . . . , a r ) be a sequence of vectors a i = (b i , c i ) in K 2 such that any pair (a i , a k ) with k = i is linearly independent, n = (n 0 , . . . , n r ) a sequence of positive integers and L = (l ij ) a family of positive integers, where 0 ≤ i ≤ r and 1 ≤ j ≤ n i . For every 0 ≤ i ≤ r, define a monomial

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“…Theorem B. If X is a noetherian, normal and excellent algebraic stack then a Cox ring of reflexive sheaves of rank one We note that our construction does not need that the Picard group (or the group of reflexive sheaves of rank one) is finitely generated, although also the classical construction in [2] does not really rely on it. Additionally the classical construction assumes that H 0 (O * X ) = * for an algebraically closed field .…”

Section: Introductionmentioning

confidence: 99%

Cox ring of an algebraic stack

Hochenegger,

Martinengo,

Tonini

2020

Preprint

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We give a proper definition of the multiplicative structure of the following rings: Cox ring of invertible sheaves on a general algebraic stack; Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish its (non-)uniqueness in terms of an Ext-group. Moreover, we compare this definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.

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Cox Rings

Cited by 23 publications

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Three lectures on Cox rings

Three lectures on Cox rings

Factorially graded rings of complexity one

Cox ring of an algebraic stack

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