Universal torsors and Cox rings

On the existence of a torsor structure for Galois covers over a complete discrete valuation ring

In this note we investigate the problem of existence of a torsor structure for Galois covers of (formal) schemes over a complete discrete valuation ring of residue characteristic $p>0$ in the case of abelian Galois groups of type (p,p,...,p).

关于正则环上挠子的若干问题（含赵逸非的附录）

Problems about torsors over regular rings (with an appendix by Yifei Zhao)

本文概述了正则环上约化群挠子的相关猜想网络，并综述了推动该问题取得进展的关键技术。

Cluster algebras are Cox rings

It was recently shown by Gross, Hacking, and Keel that, in the absence of frozen indices, a cluster A-variety with generic coefficients is the universal torsor of the corresponding cluster X-variety with corresponding coefficients. We extend this to allow for frozen vectors and corresponding partial compactifications of the A- and X-spaces. When certain assumptions are satisfied, we conclude that the theta bases of Gross-Hacking-Keel-Kontsevich give bases of global sections for every line bundle on the leaves of the partially compactified X-space. We note that our arguments work without assumi…

赋值环上扭子的Bass–Quillen猜想

The Bass--Quillen conjecture for torsors over valuation rings

本文在赋值环上对满足自然迷向性条件的约化群概形证明了仿射空间上Nisnevich扭子的下降性质，从而推广了Raghunathan定理，并将若干仿射可表示性结果从等特征推广到混合特征。

Prüfer基上的纯度与拟分裂挠子

Purity and quasi-split torsors over Prüfer bases

本研究在Prüfer环非诺特背景下证明Auslander平坦性准则与Auslander–Buchsbaum公式，推导多种群代数空间下挠子的纯度定理，解决了拟分裂约化群概形挠子的Grothendieck–Serre猜想与Nisnevich纯度猜想版本。

Extension of torsors and prime to $p$ fundamental group scheme

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_η$. Let $f: Y \rightarrow X_η$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_η$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $ψ^{(p')}: π(X_η,x_η)^{(…

Gorensteinness and iteration of Cox rings for Fano type varieties

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.

环圈 torsors 与 Abhyankar 引理

Loop torsors and Abhyankar's lemma

本文定义了在正则 Hensel 环 A 沿严格正规交叉因子 D 的局部化上的特定群概形下的环圈 torsors，给出了分类的 Galois 上同调判据，并重新审视了 Laurent 多项式环上的环圈 torsors 相关理论。

Reduction of structure for torsors over semilocal rings

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H^1(R, S) --> H^1(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H^1(X, S) --> H^1(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop al…

Effective models and extension of torsors over a discrete valuation ring of unequal characteristic

Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \to X_K. If p>2, we determine, through these invariants, when Y\to X has a structure of torsor which extends that of Y_K\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y.

通过泛环面计数具有相切条件的有理曲线的动机计数

Motivic counting of rational curves with tangency conditions via universal torsors

本文借助 Cox 环与泛 torsor 分解曲线到 Mori Dream Space 的态射模空间动机，并在环面簇情形证明满足边界相切条件的 Campana 曲线的动机版 Batyrev–Manin–Peyre 原理。

A presentation for the Cox ring of $\overline{M}_{0,6}$

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.

P-Divisors of Cox Rings

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.

Cox rings of some symplectic resolutions of quotient singularities

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.

Luna-Vust invariants of Cox rings and skeletons of spherical varieties

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…

Cox rings of K3 surfaces with Picard number two

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.

The Cox ring of an algebraic variety with torus action

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…

Cox rings of extremal rational elliptic surfaces

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.

On Cox rings of K3-surfaces

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.

关于Cox环的迭代

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。

The Cox ring of the space of complete rank two collineations

We determine the Cox ring of the space of complete rank two collineations.

Cox ring of the generic fiber

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.

Cox rings of minimal resolutions of surface quotient singularities

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.

Cox rings of almost homogeneous SL2-threefolds

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.

Equivariant Cox ring

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 …

On the Cox ring of blowing up the diagonal

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.

环上的 Albert 代数及相关 Torsor

Albert Algebras over Rings and Related Torsors

本文从几何观点利用 torsor 研究交换环上的例外 Jordan 代数，证明环上 Albert 代数允许非同构同位素且不唯一决定底层复合，与域上情形相反。

On the Cox ring of P^2 blown up in points on a line

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.

A remark on the Cox ring of $\overline{M}_{0,n}$

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 $n \ge 3$.

Cox rings and symplectic quotient singularities with torus action

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.

Finite torsors on projective schemes defined over a discrete valuation ring

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Q\to X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our the…

Computing 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…

代数栈的Cox环

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。

Cox rings of surfaces and the anticanonical Iitaka dimension

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

Let S_r be the blow-up of P^2 in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. For r <= 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S_8) is calculated up to radical. As conjectured by Batyrev and Popov, all the generating relations are quadratic.

Cox rings of du Val singularities

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

For a smooth Del Pezzo surface the direct sum of global sections of all isomorphism classes of invertible sheaves on it can be almost canonically endowed with a ring structure, called the Cox ring. We show that in characteristic 0 this ring is Gorenstein and its spectrum has rational singularities. Also a gap in math.AG/0309111 is pointed out.

Cox rings of rational surfaces and flag varieties of ADE-types

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.

The Cox ring of a spherical embedding

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.

Hilbert functions of Cox rings of del Pezzo surfaces

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

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combin…

Singularities of Cox Rings of Fano Varieties

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.

关于Calabi-Yau超曲面的Cox环

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。

Cox rings of K3 surfaces of Picard number three

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 …

嵌入变种的Cox环

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。

Log Fano structures and Cox rings of blow-ups of products of projective spaces

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.

Crepant resolutions of 3-dimensional quotient singularities via Cox rings

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.

Cox Rings

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.

Cox rings of cubic surfaces and Fano threefolds

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.

Cox Rings and Algebraic Maps

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.

作为GIT商的Cox环的半不变量环

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。

关于某些超曲面的 Cox 环

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。

Mori dream K3 surfaces of Picard number four: projective models and Cox rings

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.

通过Brown代数及相关 torsor 研究环上的$\n\thrm{E}_6$与$\n\thrm{E}_7$型群

Groups of type $\mathrm{E}_6$ and $\mathrm{E}_7$ over Rings via Brown Algebras and Related Torsors

该研究在交换环框架下分析可结构代数与Freudenthal三元系，实现E₆、E₇型例外群的对应构造，刻画6可逆时E₇/E₆型齐性空间的几何意义，并揭示环上Brown代数主同位素可存在非同构的新特性。