The structure of Galois covering spaces of the protective plane

(i)构造伽罗瓦覆盖π：X→Y对某些有限组G包含二面角组和对称组这样的伽罗瓦覆盖W→Z紧凑保护品种与加Z (W / Z)【相似或相等】G,得到的纤维产品π和理性的地图应用程序从Z Y,我们表明,对于任何伽罗瓦覆盖射影平面的伽罗瓦群同构的二面角集团订单2 r (r是奇数),对于某些齐次多项式h， g_1和g_2，分支轨迹由满足fh^2 =g^2_1 + g^r_2的齐次多项式f来定义。（ii）计算了投影平面上某些曲线的补的基本群。作为应用，我们得到了一个新的Zariski对，它的每条曲线由四条圆锥曲线组成，并且只有节点和节点作为奇点。此外，我们还给出了一种计算投影平面上伽罗瓦覆盖基群的方法。（iii）设X为投影平面的伽罗瓦覆盖，设^^-__X为X的最小分辨率，得到如下结果：如果Gal（X/P^2）的每一个子群都是正规的，则^^-__X的不规则性等于零。如果Gal（X/P^2）同构于2p阶的二面体群（P是奇素数），则^^- __x的不规则性是P - 1的倍数。（iv）我们将伽罗瓦覆盖从射影空间到它们自身进行分类。(v)给出了一种构造投影平面的伽罗瓦覆盖的新实例的方法，其通用覆盖与二维多盘或开球同构。此外，我们还展示了如何计算微分方程的线性无关的四个解的比值。

(i) We construct Galois coverings π : X → Y for certain finite groups G containing dihedral groups and symmetric groups such that any Galois covering W → Z of a compact protective variety Z with Gal(W/Z) 【similar or equal】 G, is obtained as the fiber product of π and a rational map from Z to Y. As an application, we show that for any Galois covering of the projective plane with the Galois group isomorphic to the dihedral group of order 2 r(r is odd), the branch locus is defined by a homogeneous polynomial f satisfying fh^2 =g^2_1 + g^r_2 for certain homogeneous polynomials h, g_1 and g_2.(ii) We compute the fundamental groups of the complements for certain curves on the projective plane. As an application, we obtain a new Zariski pair each curve of which consists of four conics and has only nodes and tacnodes as singularities. Moreover, we give a method computing the fundamental groups of Galois coverings of the projective plane.(iii) Let X be a Galois covering of the projective plane and let ^^-__X be the minimal resolution of X. We obtain the following results. If every subgroup of Gal(X/P^2) is normal, then the irregularity of ^^-__X is equal to zero. If Gal(X/P^2) is isomorphic to the dihedral group of order 2p (p is an odd prime), then the irregularity of ^^-__X is a multiple of p - 1.(iv) We classify Galois coverings from projective spaces to theirselves.(v) We give a method constructing new examples of Galois covering of the projective plane whose universal coverings are isomorphic to the 2-dimensional polydisk or open ball. Moreover, we show how to calculate the differential equations the ratios of whose linearly independent four solutions give uniformizations.