有限阿贝尔覆盖及其应用
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摘要
本文的主要目的是建立Abel覆盖的一般方法，并给出几个有趣的应用。
覆盖理论是代数几何研究的重要工具。要建立覆盖理论，关键是要解决以下几个基本问题：(Ⅰ)定义覆盖的简单数据的确定；(Ⅱ)正规化的计算；(Ⅲ)分歧轨迹的确定；(Ⅳ)找出奇点解消的有效方法；(Ⅴ)不变量的计算等。
二次覆盖，三次覆盖，循环覆盖都有了成熟的理论，从它们简单的定义方程出发，其相应的基本问题都得到了解决。为了代数曲面的分类和构造新曲面的需要，从上世纪90年代开始，很多代数几何学家如Pardini都希望建立Abel覆盖的基本理论。但是到目前为止，这些基本问题还未解决。Abel覆盖也没有象二次覆盖、三次覆盖和循环覆盖那样得到了有效地应用。
本文首先证明Abel覆盖有一组标准的定义方程。由此出发，计算出了Abel覆盖的正规化。因此，确定出Abel覆盖的分歧轨迹及其分歧指数；找到了奇点解消的标准方法；给出了不变量的计算公式。因此解决了Abel覆盖的上述基本问题。
利用本文所建立的Abel覆盖的方法，我们解决了下面几个有趣的问题。
第一，我们构造了82个一般型代数曲面，他们的不变量满足c_1~2=3c_2。构造这种类型的曲面是一个具有挑战性的问题．有很多知名的数学家都曾考虑过这个问题，比如：Mumford，Hirzebruch，Mostow，萧荫堂和Horikawa等。值得注意的是Hirzebruch构造的4个曲面的不规则性q的计算也是非平凡的。Ishida和Hironaka还专门写文章来计算它。而利用本文的结果，我们得到了任意Abel覆盖曲面的q的计算公式。
第二，我们对典范映射是P~2上的次数至少为4的Abel覆盖的曲面进行了分类。特别地，我们发现了4个典范次数是16的新曲面，这是目前所知道的典范映射次数最高的曲面的例子。
典范映射性态的研究是代数曲面理论中的一个困难问题。从上世纪50年代开始，很多代数几何学家研究过它，如Kodaira，Bombieri，Beauville，肖刚和Persson等。但具有高次数典范映射的曲面的存在和分类问题还仍然是一个未解决的问题。1978年，Persson构造了第一个典范映射次数为16的曲面。他的构造基于一类Campadelli曲面，而后者的构造也是非平凡的。
第三，我们给出了一个经典不等式|G|≤4g+4的新证明，这里，G是一条亏格g≥2的曲线的Abel自同构群。特别是，我们还给出了满足|G|≥4g-4的曲线的完整分类。
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 c_1~2 = 3c_2. 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 P~2 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 algebraic
geometers: 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.
覆盖理论是代数几何研究的重要工具。要建立覆盖理论，关键是要解决以下几个基本问题：(Ⅰ)定义覆盖的简单数据的确定；(Ⅱ)正规化的计算；(Ⅲ)分歧轨迹的确定；(Ⅳ)找出奇点解消的有效方法；(Ⅴ)不变量的计算等。
二次覆盖，三次覆盖，循环覆盖都有了成熟的理论，从它们简单的定义方程出发，其相应的基本问题都得到了解决。为了代数曲面的分类和构造新曲面的需要，从上世纪90年代开始，很多代数几何学家如Pardini都希望建立Abel覆盖的基本理论。但是到目前为止，这些基本问题还未解决。Abel覆盖也没有象二次覆盖、三次覆盖和循环覆盖那样得到了有效地应用。
本文首先证明Abel覆盖有一组标准的定义方程。由此出发，计算出了Abel覆盖的正规化。因此，确定出Abel覆盖的分歧轨迹及其分歧指数；找到了奇点解消的标准方法；给出了不变量的计算公式。因此解决了Abel覆盖的上述基本问题。
利用本文所建立的Abel覆盖的方法，我们解决了下面几个有趣的问题。
第一，我们构造了82个一般型代数曲面，他们的不变量满足c_1~2=3c_2。构造这种类型的曲面是一个具有挑战性的问题．有很多知名的数学家都曾考虑过这个问题，比如：Mumford，Hirzebruch，Mostow，萧荫堂和Horikawa等。值得注意的是Hirzebruch构造的4个曲面的不规则性q的计算也是非平凡的。Ishida和Hironaka还专门写文章来计算它。而利用本文的结果，我们得到了任意Abel覆盖曲面的q的计算公式。
第二，我们对典范映射是P~2上的次数至少为4的Abel覆盖的曲面进行了分类。特别地，我们发现了4个典范次数是16的新曲面，这是目前所知道的典范映射次数最高的曲面的例子。
典范映射性态的研究是代数曲面理论中的一个困难问题。从上世纪50年代开始，很多代数几何学家研究过它，如Kodaira，Bombieri，Beauville，肖刚和Persson等。但具有高次数典范映射的曲面的存在和分类问题还仍然是一个未解决的问题。1978年，Persson构造了第一个典范映射次数为16的曲面。他的构造基于一类Campadelli曲面，而后者的构造也是非平凡的。
第三，我们给出了一个经典不等式|G|≤4g+4的新证明，这里，G是一条亏格g≥2的曲线的Abel自同构群。特别是，我们还给出了满足|G|≥4g-4的曲线的完整分类。
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 c_1~2 = 3c_2. 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 P~2 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 algebraic
geometers: 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.
引文
[1] Andreotti, A.: Sopra le superficie che possegono transformazioni birazionali in se, Rend. Mat. Appl. 255-279.
[2] Beauville, A.: L'application canonique pour les surfaces de tupe general, Invent. Math. 55(1979), 121-140.
[3] Bombieri, E. and Husemoller, D. Classification and embedding of surfaces, Proceedings of Symposia in Pure Math. 39 (1974), 329-420.
[4] Barth, W., Peters, C. and Van de Ven, A.: Compact complex surfaces, SpringerVarlag Berlin Heidelberg New York Toykyo, (1984).
[5] W. Burnside. Theory of groups offinite order, Note K. Dover (1955).
[6] Catanese, F.: On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), 483-515.
[7] Catanese, F.: Singular bidouble covers and the construction of interesting algebraic surface. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97-120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, (1999).
[8] Catanese, F.: canonical ring and "special" surfaces of general type. Proc. Symp. Pure Math., 46, 175-194, Providence, RI; Am. Math. Soc. (1987).
[9] Catanese, F.: Babbage's conjecture, contact of sufgace, symmetric determinatal varieties and applications. Invent. Math. 63, 433-467, (1981).
[10] Casnati, G.: Covers of algebraic varieties Ⅱ: Covers of degree 5 and construction of surfaces. J. Algebraic Geom. 5, 461-477, (1996).
[11] Coati, A.: Polynomial bounds for the number of automorphisms of a surface of genral type. Ann. Sci. Ecole Nor. Sup. 24, 113-137 (1937)
[12] Dedekind, R.: JFM30.0198.02, J. Reine Angew. Math. 121 (1899), 40-123.
[13] P. Deligne; G. D. Mostow,: Monodromy of hypergeometric functions and non-lattice integral monodromy. Mathématiques de l'I. H. E. S, tome 63, (1986) 5-89.
[14] Esnault, H.: Fibre de Milnor d'un cone sur une courbe plane singulière. Invent. Math. 68(1982), no. 3, 477-496.
[15] Esnault, H.; Viehweg, E.: Revents cycliques. Algebraic threefolds (Varenna, 1981), pp. 241-250, Lecture Notes in Math., 947, Springer, Berlin-New York, 1982.
[16] Esnault, H.; Viehweg, E.: Revetements cycliques. Ⅱ (autour du théorème d'annulation de J. Kollár). Géométrie algébrique et applications, Ⅱ (La Rébida, 1984), 81-96, Travaux en Cours, 23, Hermann, Paris, 1987.
[17] Esnault, H.; Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86(1986), no. 1, 161-194.
[18] Esnualt, H.; Viehweg E., Lecture on vanishing theorems. DMV Seminar, 20. Birkhauser Verlag, Basel, 1992.
[19] Yun Gao, Integral closure of an abelian extension and applications to abelian covers in algebraic geometry (Submitted)
[20] van Der Geer, G.; Zagier, D.: The Hilbert modular group of the field Q[13~(1／2)], Invent. Math. 42, 93-134 (1977).
[21] Greenberg, L.: Conformal transformations of Riemann surfaces, American J. of Math. Ann. 41, (1960).
[22] R. Hartshome: Algebraic Geometry. GTM 52.
[23] W. J. Harvey: Cyclic groups of automorphisms of a compact surfaces. Quart. J. Math. Oxford(2)17, 86-97, (1996)
[24] Hironaka, E.: Abelian covering of the complex projective plane branched along configurations of Real lines, American Mathematical society, 105, (1993).
[25] Hirzebruch, F.: Arrangements of lines and algebraic surfaces. Arithmetic and geometry, Vol. Ⅱ, 113-140, Progr. Math., 36, Birkhauser, Boston, Mass., (1983).
[26] Hirzebruch, F.: Chern numbers of algebraic surfaces: an example. Math. Ann. 266(1984), no. 3, 351-356.
[28] Hirzebruch, F.: some examples of algebraic surfaces. Contemparary Mathematics. 9(1982), 55-70.
[28] Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97-120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, (1999).
[29] Horkawa, E: Algebraic surfacesw of general type with small C_1~2. V. J. Fac. Univ. Tokyo, Sect. IA 283, 745-755, (1981).
[30] Howard, A.; Sommese, A. J.: On the order of the automorphism groups of certain projective manifold. Manifolds and Lie Groups, Progress in Math. 14, 145-158,(1982).
[31] A. Hurwitz: Algebraische Gebilde mit eindeutigen transformathionen in sich. Math. Ann. 41, (1893).
[32] Huckleberry, A. T.; Sauer M.: On the order of the automosphism group of a surface of general type. Math. Z. 205, 321-329, (1990).
[33] Inoue, M.: Some surfaces of general tupe with positive indices, preprint.
[34] Ishida, M. -N.: The irregularities of Hirzebruch's examples of surfaces of general type with c_1~2= 3c_2. Math. Ann. 262, no. 3, 407-420, (1983).
[35] Lang, S.: Algebra, Addison-Wesley Publish Company, (1984).
[36] Libgober, A.: A belian branched covers of projective plane, preprint.
[37] Livné, S.: On certain covers of the universal elliptic curve, Ph. D thesis, Harvard University, (1981).
[38] C. Maclachlan: Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15, 699-712, (1981).
[39] C. Maclachlan: A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44, 265-272, (1969).
[40] R. Miranda: Triple covers in algebraic geometry, Am. J. Math. 107(5), 1123-1158, (1985).
[41] Miayaoka, Y: On the Chern numbers of surfaces of general type, Invent. Math. 42, 225-237, (1977).
[42] G. D. Mostow; Y. T. Siu: A compact Kohler surface of negative curvature not covered by the ball, Annals of Math. 112, 321-360, (1987).
[43] Schoichi Nakajima: On abelian automorphism groups of algebraic curves, J. London Math Soc. (2) 36, 23-32, (1987).
[44] Pardini, R.: Abelian covers of algebraic variety, J. Peine Angew. Math. 417 (1991), 191-213.
[45] Pardini, R.: canonical images of surfaces. J. reine angew. Math. 417, 215-219
[46] Picard, E.: Sur une extension aux fonctions de deux variables du problem de Riemann relatif aux fonctions hypergéométriques. Ann. ENS. 10, 305-322, (1881).
[47] Picard, E.: Sur les fonctions hyperfuchsiennes provenant des séries hypergéométr iques do deux variables. Ann. ENS. Ⅲ2, 357-384, (1885).
[48] Persson, U.: Double coverings and surfaces of general type. In: Olson, L. D. (ed.) Algebraic geometry. (Lect. Notes Math., vol.687, pp. 168-195). Berlin Heidelberg New York: Springer 1978.
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[50] Tan, S. -L.: Surfaces whose canonical maps are of odd degrees. Math. Ann. 292, no. 1, 13-29, (1992).
[51] Tan, S. -L.: Integral closure of a cubic extension and applications, Proe. Amer. Math. Soc. 129 no.9, 2553-2562, (2001).
[52] Tan, S. -L.: Triple covers on smooth algebraic varieties, Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics, 20(2002), 143-164.
[53] Tan, S. -L.: On the invariants of base changes of pencils of Curves, Ⅰ, manuscripta Math, 84(1994), 225-244.
[54] Tan, S. -L.: On the invariants of base changes of pencils of Curves, Ⅱ, Math. Z. 222 (1996), 665-676.
[55] Xiao G.: Algebraic surfaces with high canonical degree. Math. Ann. 274 (1986), 473-483.
[56] Xiao G.: On abelian automorphism group of a surface of general type. Invent. Math. 102(1990), 619-631.
[57] Xiao G.: Bound of automorphisms of surfaces of general type, Ⅰ. Annals of Mathematics. 139 (1994), 51-77.
[58] Xiao G.: Bound of automorphisms of surfaces of general type, Ⅱ. J. Algebraic Geom. 4(1995), 701-793.
[59] Xiao G.: Promblem list. In: Birational geometry of algebraic varieties: open problems. The 23rd International Symposium of Taniguchi Foundation, (1988), 36-41.
[60] Vasconcelos, W.: Integral closure - Rees algebras, multiplicities, algorithms. Springer Monographs in Mathematics. Springer-Vedag, Berlin, 2005.
[61] Yau, S. T.: Calabi's conjecture and some new results in albebraic geometry, Porc. Natl. Acad. Sci. USA 74 (1977).
[62] Zuo, K.: Kummer-iiberlagerungen algebraischer Flachen. Dissertation, Rheinische Friedrich-Wilhelms-Universitat, Bonn, 1988. Bonner Mathematische Schriften [Bonn Mathematical Publications], 193. Universitat Bonn, Mathematisches Institut, Bonn, 1989.
[2] Beauville, A.: L'application canonique pour les surfaces de tupe general, Invent. Math. 55(1979), 121-140.
[3] Bombieri, E. and Husemoller, D. Classification and embedding of surfaces, Proceedings of Symposia in Pure Math. 39 (1974), 329-420.
[4] Barth, W., Peters, C. and Van de Ven, A.: Compact complex surfaces, SpringerVarlag Berlin Heidelberg New York Toykyo, (1984).
[5] W. Burnside. Theory of groups offinite order, Note K. Dover (1955).
[6] Catanese, F.: On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), 483-515.
[7] Catanese, F.: Singular bidouble covers and the construction of interesting algebraic surface. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97-120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, (1999).
[8] Catanese, F.: canonical ring and "special" surfaces of general type. Proc. Symp. Pure Math., 46, 175-194, Providence, RI; Am. Math. Soc. (1987).
[9] Catanese, F.: Babbage's conjecture, contact of sufgace, symmetric determinatal varieties and applications. Invent. Math. 63, 433-467, (1981).
[10] Casnati, G.: Covers of algebraic varieties Ⅱ: Covers of degree 5 and construction of surfaces. J. Algebraic Geom. 5, 461-477, (1996).
[11] Coati, A.: Polynomial bounds for the number of automorphisms of a surface of genral type. Ann. Sci. Ecole Nor. Sup. 24, 113-137 (1937)
[12] Dedekind, R.: JFM30.0198.02, J. Reine Angew. Math. 121 (1899), 40-123.
[13] P. Deligne; G. D. Mostow,: Monodromy of hypergeometric functions and non-lattice integral monodromy. Mathématiques de l'I. H. E. S, tome 63, (1986) 5-89.
[14] Esnault, H.: Fibre de Milnor d'un cone sur une courbe plane singulière. Invent. Math. 68(1982), no. 3, 477-496.
[15] Esnault, H.; Viehweg, E.: Revents cycliques. Algebraic threefolds (Varenna, 1981), pp. 241-250, Lecture Notes in Math., 947, Springer, Berlin-New York, 1982.
[16] Esnault, H.; Viehweg, E.: Revetements cycliques. Ⅱ (autour du théorème d'annulation de J. Kollár). Géométrie algébrique et applications, Ⅱ (La Rébida, 1984), 81-96, Travaux en Cours, 23, Hermann, Paris, 1987.
[17] Esnault, H.; Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86(1986), no. 1, 161-194.
[18] Esnualt, H.; Viehweg E., Lecture on vanishing theorems. DMV Seminar, 20. Birkhauser Verlag, Basel, 1992.
[19] Yun Gao, Integral closure of an abelian extension and applications to abelian covers in algebraic geometry (Submitted)
[20] van Der Geer, G.; Zagier, D.: The Hilbert modular group of the field Q[13~(1／2)], Invent. Math. 42, 93-134 (1977).
[21] Greenberg, L.: Conformal transformations of Riemann surfaces, American J. of Math. Ann. 41, (1960).
[22] R. Hartshome: Algebraic Geometry. GTM 52.
[23] W. J. Harvey: Cyclic groups of automorphisms of a compact surfaces. Quart. J. Math. Oxford(2)17, 86-97, (1996)
[24] Hironaka, E.: Abelian covering of the complex projective plane branched along configurations of Real lines, American Mathematical society, 105, (1993).
[25] Hirzebruch, F.: Arrangements of lines and algebraic surfaces. Arithmetic and geometry, Vol. Ⅱ, 113-140, Progr. Math., 36, Birkhauser, Boston, Mass., (1983).
[26] Hirzebruch, F.: Chern numbers of algebraic surfaces: an example. Math. Ann. 266(1984), no. 3, 351-356.
[28] Hirzebruch, F.: some examples of algebraic surfaces. Contemparary Mathematics. 9(1982), 55-70.
[28] Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97-120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, (1999).
[29] Horkawa, E: Algebraic surfacesw of general type with small C_1~2. V. J. Fac. Univ. Tokyo, Sect. IA 283, 745-755, (1981).
[30] Howard, A.; Sommese, A. J.: On the order of the automorphism groups of certain projective manifold. Manifolds and Lie Groups, Progress in Math. 14, 145-158,(1982).
[31] A. Hurwitz: Algebraische Gebilde mit eindeutigen transformathionen in sich. Math. Ann. 41, (1893).
[32] Huckleberry, A. T.; Sauer M.: On the order of the automosphism group of a surface of general type. Math. Z. 205, 321-329, (1990).
[33] Inoue, M.: Some surfaces of general tupe with positive indices, preprint.
[34] Ishida, M. -N.: The irregularities of Hirzebruch's examples of surfaces of general type with c_1~2= 3c_2. Math. Ann. 262, no. 3, 407-420, (1983).
[35] Lang, S.: Algebra, Addison-Wesley Publish Company, (1984).
[36] Libgober, A.: A belian branched covers of projective plane, preprint.
[37] Livné, S.: On certain covers of the universal elliptic curve, Ph. D thesis, Harvard University, (1981).
[38] C. Maclachlan: Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15, 699-712, (1981).
[39] C. Maclachlan: A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44, 265-272, (1969).
[40] R. Miranda: Triple covers in algebraic geometry, Am. J. Math. 107(5), 1123-1158, (1985).
[41] Miayaoka, Y: On the Chern numbers of surfaces of general type, Invent. Math. 42, 225-237, (1977).
[42] G. D. Mostow; Y. T. Siu: A compact Kohler surface of negative curvature not covered by the ball, Annals of Math. 112, 321-360, (1987).
[43] Schoichi Nakajima: On abelian automorphism groups of algebraic curves, J. London Math Soc. (2) 36, 23-32, (1987).
[44] Pardini, R.: Abelian covers of algebraic variety, J. Peine Angew. Math. 417 (1991), 191-213.
[45] Pardini, R.: canonical images of surfaces. J. reine angew. Math. 417, 215-219
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