Cox rings of K3 surfaces with Picard number two
John Christian Ottem
研究Picard数2的K3曲面的Cox环的生成元和关系.特别是，我们认为考克斯环的经典例子的K3表面，如四次曲面包含一条线和椭圆K3表面。
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.
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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.
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 …
Picard-graded Betti numbers and the defining ideals of Cox rings
设X是一个光滑射影簇，其Picard群是挠自由的.我们引进复合向量空间的同源性决定的结构的最小自由分解的Cox环的X在多项式环，并显示如何这些复合物的同源性可以研究纯粹的几何方法。作为这些技巧的一个应用，我们给出了Batyrev和Popov所证明的Del Pezzo曲面（次数>1）的Cox环的一个简单的新的特征证明。
On Cox rings of K3-surfaces
研究了K3-曲面的Cox环。第一个结果是，一个K3-曲面有一个生成的Cox环当且仅当它的有效锥是多面体。此外，我们还研究了Picard数为2的K3-曲面的Cox环的生成元的度和关系，并明确计算了Picard数为2 ~ 5或作为del Pezzo曲面的双覆盖出现的具有非辛对合的一般K3-曲面的Cox环.
We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.
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.
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…
On locally factorial Fano fourfolds of Picard number two
本文分类了具有超曲面Cox环且允许三维环面有效作用的Picard数为2的局部析因Fano四重折叠。
On smooth Fano fourfolds of Picard number two
本文分类了具有一般超曲面Cox环且Picard数为2的光滑Fano四维簇。
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 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.
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.
P-Divisors of Cox Rings
对Mori Dream Space（MDS）的Cox环进行分次生成，并在除子类群上分次.因此，谱的考克斯环来与行动的代数环面的GIT商是各种问题。我们提出了相关的描述，这个考克斯环作为一个多面体因子。通过其多面体系数的形状，它将Cox环的等变结构与线性系的稳定轨迹和稳定重数的世界联系起来。
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…
The Cox ring of an algebraic variety with torus action
研究了正规完备簇X上具有代数环面作用的Cox环.我们的第一个结果将X的Cox环与X的极大几何商的Cox环联系起来。作为结果，我们得到一个完整的描述的考克斯环的生成元和关系的品种与环面行动的复杂性。此外，我们提供了一个组合的方法来考克斯环使用的语言多面体因子。应用到光滑的k*-曲面，我们的结果给出了描述的Cox环的Orlik-Wagreich图。作为例子，我们根据Klyachko的描述，显式地计算了Picard数不超过2的Gorenstein del Pezzo k*-曲面的Cox环，以及秩为2的向量丛和余切丛在环面簇上投影的Cox环.
Cox rings of extremal rational elliptic surfaces
本文确定了极值有理椭圆曲面的Cox环的一个极小生成元集。此外，我们开发了一种技术，用于计算它们之间的关系，允许在所有情况下，但三种情况下，提供一个介绍的Cox环的理想。
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
我们确定了完全秩2直射空间的Cox环。
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
本文研究了曲面商奇点的最小分解的Cox环，并给出了这类环的两种刻划。第一个是Cox环的谱方程，Cox环是仿射空间中的超曲面。第二个是一组发电机的考克斯环视为一个子环的坐标环的产品的环面和另一个表面商奇异性。此外，我们得到了一个明确的描述，最小分辨率作为一个因子在复曲面品种。
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 …
On the Cox ring of blowing up the diagonal
我们计算了Blow-ups $\mathrm{Bl}_Δ(X'\times X')$和$\mathrm{Bl}_Δ(\mathbb P_1^n)$的Cox环，其中$X'$是射影空间的乘积，$Δ$是（广义）对角线。
We introduce the notion of intrinsic Grassmannians which generalizes the well known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plücker ideal $I_{d,n}$ of the Grassmannian $\mathrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary $n$. We study their geometry and show that they satisfy Fujita's freeness conjecture.
On the Cox ring of P^2 blown up in points on a line
证明了直线上n点P^2的爆破生成Cox环.我们给出了环的显式生成元，并计算了它的关系定义理想。
A remark on the Cox ring of $\overline{M}_{0,n}$
给出了一个初等的归纳论证，证明了具有$n$标记点的稳定有理曲线的模空间$\overline{M}_{0,n}$的Cox环的某个子环对每个$n \ge 3$都是可生成的.
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。
Cox rings of surfaces and the anticanonical Iitaka dimension
本文研究了光滑射影曲面X的Cox环R（X）的有限生成与其反正则Iitaka维数k（-KX）之间的关系。
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。
Three lectures on Cox rings
2011年1月在爱丁堡举行的“Torsors：理论与应用”会议上给出的入门课程笔记。
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.
The Cox ring of a spherical embedding
设G是连通约化群，G/H是球面齐性空间.证明了G/H的G-嵌入的Cox环的生成元的自然集之间的理想关系可以通过求解某些仅依赖于齐性空间的方程得到。利用这个结果，我们描述了球面齐性空间的一些例子，使得它们的任何G-嵌入的Cox环由一个方程定义。
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.
Singularities of Cox Rings of Fano Varieties
设$X$是$\mathbb{C}$上的光滑完备Fano簇.证明了Cox环$\bigoplus_{L\in\text{Pic}(X)}H^0(X,\mathcal{O}_X(L))$是具有正则奇点的Gorenstein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。
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.
Cox Rings
这是正在建设中的介绍性文本的第一章;通过作者的网页可以获得更多的章节。我们的目的是提供一个基本的访问考克斯环及其应用在代数和算术几何。欢迎对本草案提出任何意见和建议。
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.
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。
Cox rings and pseudoeffective cones of projectivized toric vector bundles
我们研究一类特殊的环面向量丛，其中包括余切丛，其相关的Klyachko过滤是特别简单的投影。对于这些射影化的丛，我们给出了有效因子锥的生成元，并将Cox环表示为沿线性子空间序列的射影空间的爆破的Cox环上的多项式代数。作为应用，我们证明了某些环面簇的投影余切丛不是Mori梦空间，并给出了投影环面向量丛的Cox环与M_{0,n}的Cox环同构的例子.
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。
On blowing up minimal toric surfaces
本文证明了Picard秩2的极小环曲面爆破的Cox环有限生成，并给出了正规射影Q-因子曲面Cox环有限生成的充要条件。
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。
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.
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…
Cox rings of rational complexity one T-varieties
设X是一个Mori梦空间，并具有一个有效的复杂性作用环.在本文中，我们在对应于X的Cox环的仿射谱的射影线P^1的适当覆盖上构造了一个多面体因子。这种描述允许详细研究环面轨道和变形的后者。此外，我们给出了P^1的覆盖以及有限阿贝尔群A在P^1上的作用，即所谓的零度A-因子。
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