Torus embeddings and dualizing complexes
Torus embeddings and dualizing complexes
复制标题
环面嵌入和二元化复合体
DOI：
10.2748/tmj/1178229687
复制
发表时间：
1980
影响因子：
0.5
通讯作者：
Masanori Ishida
中科院分区：
文献类型：
--
作者：
Masanori Ishida
介绍。设 M 为一类光滑簇的模空间。那么，M 的最佳紧缩将是可能具有某些奇点的扩展类“简并”簇的模空间。本文的主要目的是研究什么样的奇点对于这些退化簇是合理的，而不指定任何特定类别的平滑簇。我们希望奇点足够简单，以便为平滑簇定义的不变量是可推广的，并且我们可以研究它们的一般平滑变形。在曲线的情况下，Deligne 和 Mumford [DM] 的稳定曲线理论表明，仅采用普通双点作为奇点是合理的。然而，在更高维的情况下，Oda、Seshadri 和 Ishida [OS]、[II] 的简并雅可比簇，或更一般地，Namikawa 和 Nakamura [N1]、[N2] 的稳定拟贝尔簇显示出正态交叉奇点，对于简并阿贝尔簇来说限制性太大。考虑到许多退化簇的例子，我们开始将环面嵌入作为奇点、子方案、在环面作用下不变的局部模型。因此，它们是 Mumford 等人对环形嵌入的概括。 [TE]。但这些都太笼统了，我们必须从中找出好的条件。给出局部模型为Cohen-Macaulay或Gorenstein的条件是有意义的。在平滑品种的分类中，规范可逆滑轮发挥着重要作用。塞尔对偶定理被推广到 CohenMacaulay 簇，其中规范可逆滑轮被对偶滑轮取代。如果品种是 Gorenstein，则它们是可逆的。我们稍后定义的球形度将是局部模型成为 Gorenstein 的良好条件。我们现在更详细地解释本文的内容。令 N 为秩为 r ^ 0 的自由 Z 模，并令 M 为对偶 Homz (iV, Z)。然后对于固定字段 k，仿射环面嵌入为
Introduction. Let M be the moduli space of a class of smooth varieties. Then, the best compactification of M will be the moduli space of an extended class of "degenerate" varieties which may have some singularities. The main purpose of this paper is to study what kind of singularities are reasonable for these degenerate varieties without specifying any particular class of the smooth varieties. We would like the singularities to be sufficiently simple so that invariants defined for smooth varieties are generalizable, and that we can study the generically smooth deformation of them. In the case of curves, the theory of the stable curves by Deligne and Mumford [DM] shows that it is reasonable to take only ordinary double points as the singularities. In the higher dimensional cases, however, the degenerate Jacobian varieties of Oda, Seshadri and Ishida [OS], [II] or more generally, the stable quasiabelian varieties of Namikawa and Nakamura [Nl], [N2] show normal crossing singularities to be too restrictive for the degenerate abelian varieties. Looking at many examples of degenerate varieties, we came to take, as the local models of singularities, subschemes, invariant under the torus action, of torus embeddings. Thus they are generalizations of toroidal embeddings by Mumford et al. [TE]. But these are too general, and we must find out good conditions on them. It is meaningful to give the condition for the local models to be Cohen-Macaulay or Gorenstein. In the classification of smooth varieties, the canonical invertible sheaves play an important role. The Serre duality theorem is generalized for CohenMacaulay varieties with the canonical invertible sheaves replaced by the dualizing sheaves. They are invertible if the varieties are Gorenstein. The sphericity, which we define later, will be a good condition for the local model to be Gorenstein. We now explain the content of this paper in more detail. Let N be a free Z-module of rank r ^ 0, and let M be the dual Homz (iV, Z). Then for a fixed field k, an affine torus embedding of