The structure of Galois covering spaces of the protective plane

(i)我们对包含二面体群和对称群的有限群G构造Galois覆盖π：X → Y，使得紧保护簇Z的任何Galois覆盖W → Z与Gal（W/Z）[相似或相等] G，都是π与从Z到Y的有理映射的纤维积.作为应用，我们证明了对于任意的射影平面的伽罗瓦覆盖，其伽罗瓦群同构于2 r阶二面体群（r为奇数），其分支轨迹由一个齐次多项式f定义，对于某些齐次多项式h，g_1和g_2，f满足fh ^2 =g^2_1 + g^r_2. (ii)我们计算了射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Zagliki对，它的每条曲线由四条二次曲线组成，只有节点和tacnodes作为奇点。此外，我们还给出了计算射影平面的Galois覆盖的基本群的方法。(iii)设X是射影平面的一个伽罗瓦覆盖，X的最小分解是X的最小分解。我们得到以下结果。如果Gal（X/P^2）的每个子群都是正规的，则^^-__X的无规则性等于零。如果Gal（X/P^2）同构于2 p阶二面体群（p是奇素数），则^^-_X的无规性是p - 1的倍数。(iv)我们把从射影空间到它们自身的伽罗瓦覆盖分类。(v)给出了一种构造射影平面的Galois覆盖的新例子的方法，这些覆盖的泛覆盖同构于二维多圆盘或开球。此外，我们展示了如何计算的微分方程的线性无关的四个解决方案的比率给一致化。

(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.