Multigraded factorial rings and Fano varieties with torus action
Multigraded factorial rings and Fano varieties with torus action
复制标题
具有环面作用的多级阶乘环和 Fano 簇
作者：
J. Hausen;Elaine Herppich;Hendrik Suss
在第一个结果中，我们描述了所有的代数生成的阶乘代数的特征为零的代数闭域，来与一个有效的multigradation的复杂性，通过发电机和关系。这使得我们能够通过簇的考克斯环系统地构造具有自由因子类群和复杂度为1的环面作用的簇。对于这种类型的法诺品种有一个自由的除数类组的秩1，我们提供了明确的界限可能的变形类型的数量取决于尺寸和指数的皮卡德组的除数类组。因此，可以产生固定维度和Picard指数的分类列表。我们在以下情况下进行了示范。有15个非复曲面的皮卡德指数最多为6。此外，有116个Picard指数最多为2的非环面三重;其中9个是局部阶乘，即Picard指数为1，其中1个是光滑的，6个具有正则奇点，2个具有非正则奇点。最后，有67个非复曲面局部阶乘四重和两个非复曲面局部阶乘四重的一维家庭。在所有情况下，我们明确列出了考克斯环。
In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 non-toric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly.
多面体约数和代数环面作用
影响因子：
1.4
作者：
Klaus Altmann;Jürgen Hausen
通讯作者：
Jürgen Hausen