Rigidity of quotient singularities
Rigidity of quotient singularities
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DOI：
10.1007/bf01418741
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发表时间：
1971-03
影响因子：
3.1
通讯作者：
M. Schlessinger
中科院分区：
文献类型：
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作者：
M. Schlessinger
如果有限群 G 作用于仿射格式 Y，在域 k 上平滑，具有单个不动点 yeY，则商格式 X= Y/G 在 y 下 X 中的点 x 处具有孤立奇点。我们将证明这个奇点是刚性的，只要 Y > 3，并且 G 的阶数不能被 k 的特征整除。这里的刚性意味着任何在 x 处包含 X 的胚芽的族都是平凡的；这个概念类似于马瑟和索恩理论中的稳定绘图萌芽的概念。这个刚性定理（肯定地）回答了 Andreotti 的猜想，我们用它来表明，我们不能模仿维度高于 2 的 Kummer 簇的构造，来获得具有有趣的变形理论的光滑射影簇的例子。因为，如果A是维数>2的阿贝尔簇，且G= Z/2通过a~-a（chc.4=2）作用于A，则A/G的奇点变形完全解释了A的变形理论与通过去奇异化A/G得到的Kummer簇K的变形理论之间的差异。如果 A--2 为暗淡，则 A/G 的 16 个奇异点是圆锥形的：z2= xy，具有一个变形参数族 z2= xy+ t。因此，K 比 A 具有更多模量（多 16 个），但仅限于 2 维。所使用的基本方法是将局部方案 (X, x) 的变形理论与其穿孔谱 U= Xx 的变形理论进行比较。我已经提供了一些其他刚性奇点的例子，可以通过这种方法轻松获得。 w 1. 深度变形理论 __> 3令 k 为一个域，并用 C 表示局部方案 X= Spec A 的类别，其中 A 是具有剩余类域 A/ma-k 的诺特局部 k 代数。如果 X 是 k 有理点上 k 上的有限类型方案的亨利化，我们将称 X 为几何。因此，几何局部方案的完整子类别本质上是代数方案的点胚类别，采用局部 etale 拓扑。我们将比较几何局部方案 X= Spec A 的变形理论与 U= X-x 的变形理论，x 是闭点
If a finite group G acts on an affine scheme Y, smooth over a field k, with a single fixed point yeY, then the quotient scheme X= Y/G has an isolated singularity at the point x in X under y. We shall show that this singularity is rigid, provided dim Y> 3, and the order of G is not divisible by the characteristic of k. Rigidity here means that any family containing the germ of X at x is trivial; it is a notion analogous to that of stable mapping germ in the theory of Mather and Thorn. This rigidity theorem answers (affirmatively) a conjecture of Andreotti, and we use it to show that one cannot parody the construction of Kummer varieties, in dimension higher than 2, to obtain examples of smooth projective varieties with interesting deformation theories. For, if A is an Abelian variety of dimension> 2, and G= Z/2 acts on A by a~-a (chc. 4= 2), then the deformations of the singular points of A/G account entirely for the difference between the deformation theory of A and that of the Kummer variety K obtained by desingularizing A/G. If dim A--2, then the sixteen singular points of A/G are conical: z2= xy, with one parameter family z2= xy+ t of deformations. Thus K has more moduli than A does (16 more), but only in dimension 2. The basic method used is the comparison of the deformation theory of a local scheme (X, x) with that of its punctured spectrum U= Xx. I have included a few other examples of rigid singularities, which can be obtained easily from this method. w 1. Deformation Theory in Depth __> 3Let k be a field, and denote by C the category of local schemes X= Spec A, where A is a noetherian local k algebra with residue class field A/ma-k. We shall call X geometric if it is the henselization of a scheme of finite type over k at a k-rational point. The full subcategory of geometric local schemes is thus essentially the category of point germs of algebraic schemes, taken in the local etale topology. We are going to compare the deformation theory of a geometric local scheme X= Spec A with that of U= X-x, x being the closed point