John Christian Ottem
We show that the blow-up of P^2 in n points on a line has finitely generated Cox ring. We give explicit generators for the ring and calculate its defining ideal of relations.
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We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.
We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, e.g. testing finite generation or computing an explicit presentation in terms of generators and relations. As a first application, we compute the Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show computationally that all smooth rational surfaces of Picard number at most six are Mori dream surfaces and we provide explicit presentations of the Cox ring for those not admitting a torus action. Finally, we provide the Cox…
We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.
We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.
We compute the relations in the Cox ring of the moduli space $\overline{M}_{0,6}$. This gives a presentation of the Cox ring as a quotient of a polynomial ring with 40 generators by an ideal with 225 generators that come in 5 symmetry classes.
The Cox ring of a so-called Mori Dream Space (MDS) is finitely generated and it is graded over the divisor class group. Hence the spectrum of the Cox ring comes with an action of an algebraic torus whose GIT quotient is the variety in question. We present the associated description of this Cox ring as a polyhedral divisor. Via the shape of its polyhedral coefficients, it connects the equivariant structure of the Cox ring with the world of stable loci and stable multiplicities of linear systems.
We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition concerning monomial valuations it is sufficient. Also, three 4-dimensional examples are described in detail. Generators of the (expected) Cox rings of symplectic resolutions are computed and in one case a resolution is constructed as a GIT quotient of the spectrum of the Cox ring.
Given the Luna-Vust invariants of a spherical variety, we determine the Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce an explicit description of the divisor class group of the Cox ring. It follows that every spherical variety admits iteration of Cox rings. Moreover, we obtain a combinatorial proof of the fact that the Cox ring is determined by the spherical skeleton, which is a known result following from the description of the Cox ring due to Brion. Finally, we show that a conjectural combinatorial smoothness criterion can be reduced to the case of a factorial…
We study generators and relations of Cox rings of K3 surfaces of Picard number two. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and elliptic K3 surfaces.
We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth k*-surfaces, our results give a description of the Cox ring in terms of Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings o…
In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to provide a presentation of the Cox ring.
We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.
On iteration of Cox rings
本文刻画了所有具有复杂度为一的环面作用且容许Cox环迭代的代数簇。
We determine the Cox ring of the space of complete rank two collineations.
Given a surjective morphism $π\colon X\to Y$ of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of $π$ from the Cox ring of $X$. As a corollary we show that in some cases it is also possible to recover the Cox ring of a very general fiber, and finally we give an application in the case of the blowing-up of a toric fiber space.
We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.
We study Cox rings of normal threefolds on which SL2 acts with a dense orbit. Exploiting the method of U-invariants, we obtain combinatorial criteria for the total coordinate space and the base variety to have log terminal singularities. Then, we investigate the iteration of Cox rings and obtain a bound on its length. Finally, we develop a general approach to the description of the Cox ring by generators and relations which is effective for normal SL2 /$μ$n-embeddings.
Universal torsors and Cox rings
研究了有理曲面上的泛扭体方程。
We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation by generators and relations of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The …
We compute the Cox rings of the blow-ups $\mathrm{Bl}_Δ(X'\times X')$ and $\mathrm{Bl}_Δ(\mathbb P_1^n)$ where $X'$ is a product of projective spaces and $Δ$ is the (generalised) diagonal.
We present an elementary inductive argument proving that a certain subring of the Cox ring of the moduli space $\overline{M}_{0,n}$ of stable rational curves with $n$ marked points is finitely generated for every $n \ge 3$.
We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
Cox rings of algebraic stacks
本文严格定义了一般代数栈及正规优秀代数栈上的Cox环乘法结构，证明了其存在性及基于Ext群的（非）唯一性，并与簇上经典构造对比，应用于Mori dream stack理论。
In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension k(-K_X).
On the Cox ring of Del Pezzo surfaces
设S_r是P^2在r个一般点上的爆破，即，9-r度的光滑Del Pezzo表面。对于r <= 7，我们确定了明确定义其Cox环的二次方程.本文计算了Cox（S_8）中关系的理想，直到根式。正如Batyrev和Popov所指出的，所有的生成关系都是二次的。
Notes of an introductory course given at the conference "Torsors: Theory and Applications" in Edinburgh, January 2011.
In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities $\mathbb{D}_n,\mathbb{E}_6,\mathbb{E}_7$ and $\mathbb{E}_8$.
The Cox Ring of a Del Pezzo Surface Has Rational Singularities
对于光滑的Del Pezzo曲面，其上所有可逆层同构类的整体瓣的直和几乎可以正则地被赋予环结构，称为Cox环。我们证明了在特征0中，这个环是Gorenstein环，并且它的谱具有有理奇点。还指出了AG/0309111在数学方面的差距。
The Cox rings of del Pezzo surfaces are closely related to the Lie groups E_n. In this paper, we generalize the definition of Cox rings to G- surfaces defined by us earlier, where the Lie groups G=A_n, D_n or E_n. We show that the Cox ring of a G-surface S is almost determined by an irreducible representation V of G, and is generated by degree one elements. The Proj of this ring is a sub-variety of the orbit of the highest weight vector in V, and both are closed sub-varieties of the projective space P(V) defined by quadratic equations.
Let G be a connected reductive group and G/H a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of G/H can be obtained by homogenizing certain equations which depend only on the homogeneous space. Using this result, we describe some examples of spherical homogeneous spaces such that the Cox ring of any of their G-embeddings is defined by one equation.
To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green-Lazarsfeld indices, the projective dimensions, and the Castelnuovo-Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four.
Cox rings and combinatorics II
我们研究了具有一个非线性生成的Cox环的品种。在第一部分中，我们推广了一个组合的方法，在早期的工作中的品种与无挠的除子类群扭转的情况下。然后我们转向修改，例如，爆炸，以及考克斯环在这样的地图下如何变化的问题。我们回答这个问题的某一类修改引起的修改环境复曲面品种。此外，我们还证明了每一个具有双生成Cox环的簇都可以从一个组合最小的环在复曲面环境修改的有限级数中显式地构造出来.
Let $X$ be a smooth complete Fano variety over $\mathbb{C}$. We show that the Cox ring $\bigoplus_{L\in\text{Pic}(X)}H^0(X,\mathcal{O}_X(L))$ is Gorenstein with canonical singularities.
On Cox Rings of Calabi-Yau hypersurfaces
本文研究了平滑反canonical Calabi-Yau超曲面在平滑的toric Fano变种中的Cox环，通过研究环境Fano多面体的原始对的组合学以及嵌入变种的Cox环描述，确定了几种配置使得超曲面是Mori梦想空间，并获得了其Cox环的显式表示。此外，本文还展示了某些组合配置迫使birational自同构群为无限，从而在三维和四维中揭示了Cox环有限生成与无限birational自同构群之间的二元对立。最后，对于一类非Mori梦想的例子，证明了Morrison-Kawamata锥猜想对于可动锥。
Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox ring $R(X)$ has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrations with $f\cdot f'=2$. This result and techniques using Koszul's type exact sequences allow to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with …
The Cox ring of an embedded variety
本文研究了在 Mori 梦想空间中嵌入的变种 X 的 Cox 环，假设拉回映射在除子类群水平上诱导同构。通过证明 X 的 Cox 环是 Z 的 Cox 环商像的有限多个局部化的交集，从而给出一个终止当且仅当 X 的 Cox 环有限生成的算法，推广了先前的相关工作，并应用于计算光滑的 toric 变种中的超曲面的 Cox 环。
The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.
We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions and relations between them. In particular, we treat explicitly several examples where $G$ contains elements of age 2.
This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages. Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry. Any comments and suggestions on this draft will be highly appreciated.
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.
Given a morphism $F : X \rightarrow Y$ from a Mori Dream Space $X$ to a smooth Mori Dream Space $Y$ and quasicoherent sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ , we describe the inverse image of $\mathcal{G}$ by $F$ and the direct image of $\mathcal{F}$ by $F$ in terms of the corresponding modules over the Cox rings graded in the class groups.
The semi-invariant ring as the Cox ring of a GIT quotient
该研究针对满足特定线性化映射同构条件的GIT商，证明其Cox环同构于底空间中θ-稳定轨迹的半不变量环，该结论可应用于多类代数簇，并给出仿射平面n点Hilbert概形Cox环的显式计算。
On the Cox rings of some hypersurfaces
本文引入一种更直接的上同调方法计算 P^1 x P^n 中超曲面的 Cox 环，证明由正则序列定义的光滑超曲面是 Mori dream 空间，并具体计算了双次数 (2,4) 的 Calabi--Yau 三维簇的 Cox 环，揭示了其在光滑族中的不连续跳跃现象。
In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose Néron-Severi lattices have been classified by È.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space.
We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomor…
We study a toric degeneration of the Cox ring of the moduli of principal $SL_m(\mathbb{C})$ bundles on the projective line, with quasi parabolic data given by the the stabilizer of the highest weight vector in $\mathbb{C}^m$ and its dual $\bigwedge^{m-1}(\mathbb{C}^m)$. The affine semigroup algebra resulting from this degeneration is described using the $K-$Pieri rule from Kac-Moody representation theory. Along the way we give a proof of the $K-$Pieri rule which utilizes the classical Pieri rule and elements of commutative algebra, and we describe a relationship between the Cox ring and a clas…
We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other …
Theta Function Basis of the Cox ring of Postive 2d Looijenga pairs
本文构造了带仿射内部的二维 Looijenga 对之 Cox 环的典范 theta 函数基，并用镜像万有形变空间中的 k-解析圆盘计数解释其结构常数。
Cox rings and combinatorics
对于一个簇的一个新的生成的总坐标环，我们描述的基本几何性质的某些组合结构生活在其除子类群。例如，我们描述的奇异性，我们计算的充分锥，我们给简单的法诺准则。正如我们通过几个例子所示，这些结果允许显式计算。
Geometric Invariant Theory via Cox Rings
我们考虑了约化群在一类具有非生成Cox环的簇上的作用，例如，在射影空间的乘积上的对角作用的经典情形。给定这样一个行动，我们通过组合数据的Cox环的所有最大的开子集，使商是quasiprojective或嵌入到一个环面品种。作为应用，我们得到了一个明确的描述的腔室结构的线性化的样本锥和几个Gelfand-MacPherson型对应关系约化群的代数与环面作用的代数。此外，我们的方法提供了许多由此产生的商空间的几何信息。
Iteration of Cox rings of klt singularities
本文定义了klt奇点的Cox环迭代，证明其在$k\geq c(n)$时稳定，并利用Cox环确立了scfc覆盖的存在性及其支配性质，同时给出了复杂度为一的klt奇点的完整描述，且将相关定理推广至Fano type morphisms。
We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification $X$ of $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to obtain a homogeneous ideal in the Cox ring of $X$ and generalize the eigenvalue, eigenvector theorem for rootfinding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate system…
We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.
Let X be a Mori dream space together with an effective torus action of complexity one. In this note, we construct a polyhedral divisor on a suitable covering of the projective line P^1 which corresponds to the affine spectrum of the Cox ring of X. This description allows for a detailed study of torus orbits and deformations of the latter. Moreover, we present coverings of P^1 together with an action of a finite abelian group A in terms of so-called A-divisors of degree zero on P^1.
The Cox ring of a Del Pezzo surface
设$X_r$是由$¶^2$通过在一般位置上炸开$r \leq 8$点而得到的光滑Del Pezzo曲面。众所周知，对于$r \in \{3,4,5,6,7,8 \}$，Picard群$\Pic(X_r)$包含规范根系统$R_r \in \{A_2 \times A_1, A_4, D_5, E_6, E_7, E_8 \}$。本文证明了$X_r$（$r \geq 4$）的Cox环的一些一般性质，并证明了它与根系统$R_r$的代数群$G$的某些不可约表示中最高权向量的轨道的齐次坐标环的相似性.
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric arieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.
We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent co…
In this paper we work with a series whose coefficients are the Euler characteristic of Chow varieties of a given projective variety. For varieties where the Cox ring is defined, it is easy to see that in this case the ring associated to the series is the Cox ring. If this ring is noetherian then the series is rational. It is an open question whether the converse holds. In this paper we give an example showing the converse fails. However we conjecture that it holds when the variety is rationally connected. As an evidence of this conjecture, It is proved that the series is not rational, and in a…
Computing Cox rings via the cone conjecture
该研究利用莫里森 - 川又梦空间概念，为森梦空间建立类似胡 - 基尔GIT构造，通过对应关系和代数性质研究考克斯环，给出特定超曲面考克斯环显式表示并证明其为稠密\(F\)-纯型。
Log terminal singularities, platonic tuples and iteration of Cox rings
本文将曲面奇点作为有限可解群商并对应 Cox 环迭代的图景推广到任意维数、复杂度为一的环面作用情形，并对复合 du Val 三维奇点给出完整分类及所有迭代 Cox 环链。
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