Finite Abelian Covers And Applications

The main purpose of this thesis is to develop a general method of abelian covers and to give some interesting applications.The theory of finite covers is an important tool in algebraic geometry. In order to establish the theory of finite covers, the crucial points are to solve the following basic problems: (I) find out the defining data of the covers; (II) compute the normalization of the covers; (III) determine the branch locus; (IV) give an effective method to resolve the singularities; (V) compute the basic invariants and so on.Double covers, triple covers and cyclic covers are well understood, and the basic problems have been solved starting from the simple defining equations. In order to construct and classify algebraic surfaces, some algebraic geometers, such as R. Pardini, have tried to establish the basic theory of abelian covers since 1990s. However, up to now, the basic problems haven't been solved yet, and abelian covers have not been used as effectively as double covers, triple covers and cyclic covers.In this thesis, we prove first that any abelian cover can be defined by some standard equations. Then we find out the normalization starting from the defining equations. As a consequence, we determine the branch locus and its ramification index, we give a standard method to resolve the singularities, and we get the formulas for the computation of the basic invariants. Therefore, the problems from (I) to (V) are solved.As applications of our method, we solved several interesting problems as follows.Firstly, we construct 82 algebraic surfaces of general type with c₁² = 3c₂. To construct such surfaces is a challenging problem, which attracted the attention of many distinguished mathematicians, e.g., Mumford, Hirzebruch, Mostow, Y.T. Siu, Horikawa and so on. Note that the computation of the irregularity q of Hirzebruch's 4 surfaces is highly non-trivial. Actually, Ishida and Hironaka published papers on this computation. As an application of our results, we obtain a formula for the computation of the irregularity q.Secondly, we classify surfaces whose canonical maps are abelian covers on P² with degree at least 4. Especially, we found 4 surfaces whose canonical maps are of degree 16, which are the surfaces with highest canonical degree we know up to now.In the theory of algebraic surfaces, a difficult problem is to understand the behavior of the canonical map. Since 1950s, this problem has been studied by many algebraicgeometers: Kodaira, Bombieri, Beauville, Gang Xiao, Persson, etc. It is still an open problem whether there exist surfaces with high canonical degree. In 1980, Persson constructed the first surface whose canonical map is of degree 16, his construction is based on a kind of Campadelli surfaces whose construction is non-trivial.Thirdly, we found a new proof of the classical inequality |G| ≤ 4g + 4 for the automorphism group G of a curve of genus g ≥ 2. Furthermore, we give a complete classification of the curves with |G| ≥ 4g — 4.