Research on the structures of hypersurfaces and their function fields

设V是P^<n+1>中的光滑超曲面。我们考虑一个V从P∈P^<n+1>到超平面H的投影，这个投影导出一个域k(V)/k(H)的扩展，它不依赖于H的选择，如果扩展是伽罗瓦点，则P点称为伽罗瓦点。如果P∈V [P .]P[不是V的成员]，则称P为内域。外]伽罗瓦点。我们用δ(V)表示。δ(V^c)]；外]伽罗瓦点。从几何的角度研究了扩展K/K_P，特别考虑了以下问题：(1)求出所有的伽罗瓦点。对于分数的分配有什么规则吗？(2)求各点P∈P^<n+1>处伽罗瓦群G_P的结构。(3)求出L_P的非奇异投影模型的结构。结果表明：如果V在d大于等于4的超曲面中是一般的，那么它没有伽罗瓦点。若d = 4且d[大于等于]5，则δ(V)[小于等于]4（[n/2] + 1）和δ(V)[小于等于][n/2] + 1。另一方面，我们有δ(V^c)小于等于n + 2。当且仅当V投影等价于费马量时，等式成立。

Let V be a smooth hypersurface in P^<n+1>. We consider a projection of V from P ∈ P^<n+1> to a hyperplane H. This projection induces an extension of fields k(V)/k(H), which does not depend on the choice of H. The point P is called a Galois point if the extension is Galois. If, moreover, P ∈ V [resp. P 【not a member of】 V], then we call P an inner [resp. outer] Galois point. We denote by δ(V) [resp. δ(V^c)] the number of inner [resp. outer] Galois points. We have studied the extension K/K_P from geometrical points of view, especially we have considered the following problems:(1) Find all the Galois points. Do there exist any rules for the distribution of the points ?(2) Find the structure of the Galois group G_P at each point P ∈ P^<n+1>.(3) Find the structure of a nonsingular projective model of L_P.As results we have obtained the following; If V is general in the class of hypersurfaces with d 【greater than or equal】 4, then it has no Galois point. If d = 4 and d 【greater than or equal】 5, then δ(V) 【less than or equal】 4([n/2] + 1) and δ(V) 【less than or equal】 [n/2] + 1 respectively. On the other hand we have δ(V^c) 【less than or equal】 n + 2. The equality holds true if and only if V is projectively equivalent to the Fermat variety.