Orbits on Linear Algebraic Groups
Orbits on Linear Algebraic Groups
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线性代数群上的轨道
DOI：
10.2307/1970884
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发表时间：
1971
期刊：
影响因子：
--
通讯作者：
D. Birkes
中科院分区：
文献类型：
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作者：
D. Birkes
设G是一个线性代数群，p：GGL(V)是G的有理表示，当G是线性约化的时，D.Mumford证明了如果一个点xCV在其轨道的Zariski闭包CL(Gsx)中有0，则存在一个单参数子群X：GmG使得x(A)x0为0(定理(4.1))。(符号和定义见？2。)假设G和p：定义在域k上，且x C Vk。Mumford猜想(基于J.Tits的一个更强的猜想见[13，p.])，当k是完全时，X可以被选择在k上定义。更一般地，人们可以问一个线性代数k-群G何时具有下列性质：
Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x) of its orbit, then there exists a one-parameter subgroup X: Gm G such that x(a) x 0 as a 0 (Theorem (4.1)). (See ? 2 for notation and definitions.) Suppose that G and p :are defined over a field k and that x C Vk. It has been conjectured by Mumford (based on a stronger conjecture of J. Tits see [13, p. 64]) that, when k is perfect, X can be chosen to be defined over k. More generally, one can ask when a linear algebraic k-group G has the following property: