Complex Analysis and Algebraic Geometry: Coverings of the Rational Double Points in Characteristic p
Complex Analysis and Algebraic Geometry: Coverings of the Rational Double Points in Characteristic p
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复分析与代数几何：特征p中有理双点的覆盖
DOI：
10.1017/cbo9780511569197.003
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发表时间：
1977
影响因子：
0.2
通讯作者：
M. Artin
中科院分区：
文献类型：
--
作者：
M. Artin
特征为零的曲面的有理二重点与SL2[6，7]的有限子群G有关。也就是说，如果V表示具有线性G-作用的仿射平面，则簇X=V/G在原点有一个奇点，该奇点对应于G。设p为素数。如果p除以G的阶时，这个子群将在模p被约化时退化，而V的光滑约化通常与X的等奇约化不相容。然而，特征p中的每个有理双点都有一个光滑方案的有限(可能分支)覆盖。本文用直接计算的方法证明了这种覆盖的存在性，并计算了奇点的局部基本群。
The rational double points of surfaces in characteristic zero are related to the finite subgroups G of SL2 [6, 7]. Namely, if V denotes the affine plane with its linear G-action, then the variety X= V/G has a singularity at the origin, which is the one corresponding to G. Let p be a prime integer. If p divides the order of G, this subgroup will degenerate when reduced modulo p, and the smooth reduction of V will usually not be compatible with an equisingular reduction of X. Nevertheless, it turns out that every rational double point in characteristic p has a finite (possibly ramified) covering by a smooth scheme. In this paper we prove the existence of such a covering by direct calculation, and we compute the local fundamental groups of the singularities.