Am application of Kempf complex to hyper-discriminant

[3环定义的对称群S_n上的偏序]我们定义了n次对称群S_n上的偏序（x <等于）y, if y=a_1（三键）a_kx, i(y)=i(x)+2k，其中a_1，【三键】，a_k是连续增加或减少三个字母的3环，i（*）是S_n的元素*逆的个数，基于弱Bruhat序的类比。在此顺序中，是否考虑偶异与恒等相比较。证明了所有将1映射到n或n-1的n次偶排列都与恒等可比较。设f: V→V是V=λ型向量空间V的一个幂零线性变换，即。Jordan嵌段的大小λ_1【大于等于】λ_2【大于等于】【三键】【大于等于】λ_ι。对于f稳定子空间W (V)f(W)∧W， W和V/W的类型分别为f诱导的映射f| W: W→W和fv/ W: V/W→V/W的类型。对于划分ν和μ，我们研究集合S(λ,ν,μ)={W∧V；f(W)∧W，型W=ν，型V/W=μ}， S（λ,ν,μ）的Zariski闭包X（λ,ν,μ）在维数为|ν|的V子空间的格拉曼空间中的奇异轨迹。证明了S（λ,ν,μ）是非奇异的，其连通分量是有理数；引入一般向量，定义了X（λ,ν,μ）不可约分量的一般点，这些不可约分量的plumker坐标非常简单，可以表示它们的定义方程。我们明确地描述了具有余维2的奇异轨迹的仿射开集X（λ,(d),μ）的坐标环。

[A partila order on the symmetric groups defined by 3-cycles] We define a partial order on the symmetric group S_n of degree n by x【less than or equal】y iff y=a_1【triple bond】a_kx with i(y)=i(x)+2k where a_1,【triple bond】, a_k are 3-cycles of increasing or decreasing consecutive three letters and i(*) is the nmber of inversions of the element * of S_n, on the analogy of the weak Bruhat Order. Whether an even parmutaion is comparable to the identity or not in this ordering is considered. It is shown that all of the even permutations of degree n which map 1 to n or n-1 are comparable to the identity.[The varieties of subspaces stable under a nilpotent transformation] Let f : V → V be a nilpotent linear transformation of a vector space V of type V=λ,i.e. the size of Jordan blocks λ_1【greater than or equal】λ_2【greater than or equal】【triple bond】【greater than or equal】λ_ι. For an f-stable subspace W of V,i.e. f(W) ⊂ W, the types of W and V/W are those of the maps f|w : W → W and fv/w : V/W → V/W induced by f, respectively. For partitions ν and μ we investigate the set S(λ,ν,μ)={W ⊂ V;f(W) ⊂ W,type W=ν,type V/W=μ} and the singular locus of the Zariski closure X(λ,ν,μ) of S(λ,ν,μ) in the grassmaniann of subspaces of V of dimension |ν|. We show that S(λ,ν,μ) is nonsingular and its connected components are rational varieties ; generic vectors are introduced, which define the generic points of the irreducible components of X(λ,ν,μ) whose Plucker coordinates are fairly simple to express their defining equations. We describe explicitly the coordinate ring of an affine openset of X(λ,(d),μ) with the singular locus of codimension two.