论文标题
模棱两可的考克斯环
Equivariant Cox ring
论文作者
论文摘要
我们定义了具有代数群动作的正常品种的均等COX环。我们研究该对象的代数和几何方面，并显示其与普通Cox环的相关性。然后，我们专注于在连接的还原组G的作用下具有正常理性的复杂性变种的情况。我们表明，G-均衡性Cox环是有限生成的积分正常G-Algebra。在轻度的额外条件下，我们由发电机及其U-Invariants的子代数的关系进行了介绍，在那里您是G的Borel子组的一部分。普通的Cox环也有限地生成，并且在u-Equivariant Cox环上也是有限的同构，因此它继承了U-Alge u-alge of-alge of-alge of-alge cox rapla。我们依靠Hausen和Herphich的作品，证明Cox环的U-Invariants的子代数是有限生成的Cox环，该环在圆环的作用下是各种复杂性。这尤其表明后一个代数是一个完整的交叉点。将有限生成的Cox环中的奇异点的对数终端递送是一个有趣的问题，特别是因为通过Cox环的奇异性（[13]），Gongyo，Okawa，Sannai，Sannai和Takagi的作品表征了Fano类型的品种。我们为Cox环提供了组合性质的标准，即复杂性的几乎均匀的g变化，具有对数末端奇异性。Cox环的术语是由Arzhantsev，Braun，Hausen，Hausen和Wrobel在[1]中引入的。对于具有复杂性的圆环动作的对数末端的准酮，他们证明了迭代序列是有限的，具有有限生成的阶乘大师Cox环。我们证明，迭代序列对于正常有理的g-varieties的等效性和普通的Cox环是有限的，它满足了温和的额外条件（例如，完整品种或几乎均匀的品种）。在几乎均匀的情况下，我们证明了e夫和普通的大师考克斯环有限生成和阶乘。
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 ordinary Cox ring is also finitely generated and canonically isomorphic to the U-equivariant Cox ring, so that it inherits a canonical structure of U-algebra. Relying on a work of Hausen and Herppich, we prove that the subalgebra of U-invariants of the Cox ring is a finitely generated Cox ring of a variety of complexity one under the action of a torus. This yields in particular that this latter algebra is a complete intersection.Characterizing the log terminality of singularities in a finitely generated Cox ring is an interesting question, particularly since the work of Gongyo, Okawa, Sannai and Takagi characterizing varieties of Fano type via singularities of Cox rings ([13]). We provide a criterion of combinatorial nature for the Cox ring of an almost homogeneous G-variety of complexity one to have log terminal singularities.Iteration of Cox rings has been introduced by Arzhantsev, Braun, Hausen and Wrobel in [1]. For log terminal quasicones with a torus action of complexity one, they proved that the iteration sequence is finite with a finitely generated factorial master Cox ring. We prove that the iteration sequence is finite for equivariant and ordinary Cox rings of normal rational G-varieties of complexity one satisfying a mild additional condition (e.g. complete varieties or almost homogeneous varieties). In the almost homogeneous case, we prove that the equivariant and ordinary master Cox rings are finitely generated and factorial.
