Research on the structures of hypersurfaces and their function fields

设V是P ^<n +1>中的光滑超曲面.考虑V从P ∈ P ^<n +1>到超平面H的投影.这个投影导致了域k（V）/k（H）的扩张，它不依赖于H的选择。点P称为伽罗瓦点，如果扩张是伽罗瓦。如果，而且，P ∈ V [resp.我们称之为“P”，即“P”。伽罗瓦点。我们用δ（V）[resp. [001 pdf 1st-31files] δ（V ^c）]内部[resp.外]伽罗瓦点本文从几何的角度研究了扩张K/K_P，特别是考虑了以下问题：（1）求所有的Galois点。分数的分配有什么规则吗？(2)求伽罗瓦群G_P在P ∈ P ^<n +1>的每一点处的结构。(3)给出了L_P的一个非奇异投射模型的结构，得到了如下结果：如果V在d ≥ 4的超曲面类中是一般的，则V没有Galois点。若d = 4且d [大于或等于] 5，则δ（V）[小于或等于] 4（[n/2]+1）和δ（V）[小于或等于][n/2]+1分别成立。另一方面，我们有δ（V ^c）[小于或等于] n +2。等式成立当且仅当V射影等价于费马簇。

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.