Research on space curve and its Galois line

设C、L和L_0分别为投影三维空间P^3中的曲线和直线。考虑一个以L为中心的投影p_L:P^3—→L_0，其中L与L_0没有交集。将p_L限定为C，得到了一个态射p_L|C:C—→L_0和域的扩展：k(C)/k（L_0）。我们研究了c的扩展的代数结构和几何结构。如果这个扩展是伽罗瓦，那么我们称L为伽罗瓦线。特别地，我们研究了伽罗瓦群的结构和伽罗瓦线的数目，对于一些特殊情况，例如，当C的阶为素数时，我们得到了伽罗瓦线的数目最多为1。在完成了第一个目标之后，我们开始进行以下的研究：设V为光滑投影变分，D为非常充裕的除数。设f:V—→P^N为与|D|相关的投影嵌入。考虑一个以W为中心的投影p，使得模糊的W= n- n-1和f(V)不满足W。如果p f:V—→p ^n引起函数域的伽罗瓦扩展，则称（V，D）定义了伽罗瓦嵌入。在此条件下，我们给出了盖层伽罗瓦群的几个性质。在一般性讨论之后，我们对阿贝尔曲面进行了详细的研究。

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.