The structure of Galois covering spaces of the protective plane

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(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.