Study on Galois embeddings of algebraic surfaces

我们研究了代数曲线曲面的Galois嵌入，特别是有理曲线和交换曲面的Galois嵌入.在交换曲面的情况下，我们得到了Galois群的所有可能的类型，它们可以表现为覆盖变换群.此外，我们还列举了许多具有给定Galois群嵌入的阿贝尔曲面的例子。特别地，我们证明了这样的阿贝尔曲面与两条椭圆曲线的乘积同源。另一方面，我们找到了最少的N使得阿贝利南曲面有嵌入到PAN中。结合这一研究，我们对奇异平面有理曲线进行了研究。我们确定了伽罗华群的所有可能类型，即循环、二面体、A_4、S_4和S_4，并给出了这些伽罗瓦群的例子。结合这一研究，我们研究了伽罗瓦自同构是否可以推广到二次变换。结果我们得到了许多有理曲线，使得自同态不能推广到二次变换，例如，我们发现了只有结点作为奇点的有理曲线，且有外Galois点的次数大于7。

We have studied the Galois embedding of algebraic curves and surfaces, especially rational curves and abelian surfaces.In the case of abelian surfaces we have obtained all the possible types of the Galois groups which can appear as the covering transformation groups. Moreove we listed a lot of examples of abelian surfaces with given Galois groups of embeddings. In particular we have shown that such abelian surfaces are isogenous to the products of two elliptic curves. On the other hand, we have found the least number N such that abelinan surfaces have the embeddings into PAN. Concernig this study we have studied for singular plane rational curves. We determined all possible type of Galois group, i.e., they are cyclic, dihedral, A_4, S_4 and S_4 and have shown the examples with such Galois groups. Connecting with this research, we have studied if the Galois automorphism can be extended to a birational transformation or not. As a result we have obtained that there are a lot of rational curves such that the autopmorphism cannot be extended to birational transformation, for example we found rational curve with only nodes as singularities and the degree is bigger than 7 with outer Galois point.