Research on space curve and its Galois line

设C、L和L_0分别是射影空间P^3中的一条曲线和一条直线。考虑以L为中心的投影p_L：p^3--→L_0，其中L和L_0不相交。将p_L限制到C，得到一个态射p_L|C：C--→L_0和域的扩张：k(C)/k(L_0)。我们研究了C的扩张的代数结构和几何结构。如果这个扩张是伽罗华线，那么我们称L为伽罗瓦线。特别地，我们研究了一些特殊情况下的伽罗瓦群的结构和伽罗瓦线的个数，例如，如果C的次数是素数，则这个数至多为1。在完成第一个目标后，我们开始研究如下研究：设V是光滑射影簇，D是一个非常充分的因子。设f：v--→P^N是与|D|相关的射影嵌入。考虑一个以W为中心的投影p，使得dim W=N-n-1且f(V)不满足W。如果f：V-→P^n引出函数域的伽罗瓦扩张，则称(V，D)定义了伽罗瓦嵌入。在此条件下，我们证明了覆盖的Galois群的几个性质。在一般的讨论之后，我们详细地研究了阿贝尔曲面的主题。

Let C, L and L_0 be a curve and lines in the projective three space P^3 respectively. Consider a projection p_L:P^3--→L_0 with center L, where L and L_0 have no intersection. Restricting p_L to C, we get a morphism p_L|C:C--→L_0 and an extension of fields:k(C)/k(L_0). We have studied the algebraic structure of the extension and the geometric one of C. If this extension is Galois, then we call L a Galois line. In particular we have studied the structure of the Galois group and the number of Galois lines for some special cases, for example, we obtained that the number is at most one if the degree of C is a prime number. After completed the first aims, we started to study the following research : Let V be a smooth projective variety and D be a very ample divisor. Let f:V--→ P^N be the projective embedding associated with |D|. Consider a projection p with a center W such that dim W=N-n-1 and f(V) does not meet W. Ifp f:V--→ P^n induces a Galois extension of function fields, then (V,D) is said to define a Galois embedding. Under this condition we have shown several properties of the Galois group of the covering. After general discussions we study the subject for abelian surfaces in detail.