On weierstrass points and automorphisms of curves of genus three
On weierstrass points and automorphisms of curves of genus three
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论三亏格曲线的韦尔斯特拉斯点和自同构
DOI：
10.1007/bfb0066649
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
Kaname Komiya
中科院分区：
文献类型：
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作者：
Akikazu Kuribayashi;Kaname Komiya
本文的目的是给出一个3属曲线的自同构群G。设H是G的最大阶的循环子群，设n=# G是G的阶，设m=# H是H的阶，在§i中我们研究了黎曼球面的循环覆盖的黎曼曲面。如果循环群的m阶大于4，则主要利用经典而有力的Riemann-Hurwitz关系得到定理1。在§2和§3中，我们研究了由y3= x (xl)(x-tl)（x-t2）和y4= x (xl)（xt）定义的黎曼曲面。通过考虑这些黎曼曲面的Weierstrass点和它们在~ 2中的正则模型，得到了定理2和定理3。在§4中，我们考虑用weerstrass点表征的特殊黎曼曲面。我们具体地得到了这些黎曼曲面的自同构群和用来定义这些曲面的方程。在§5中我们研究了黎曼曲面，它是复环面的循环覆盖。我们得到了定理4，其中提出了一些新的曲线。主要结果如下：
The aim of this paper is to give the group G of automorphisms of a curve of genus three. Let H be the cyclic subgroup of maximum order of G. Let n=# G be the order of G and let m=# H be the order of H. In § i we study Riemann surfaces which are cyclic coverings of the Riemann sphere. If the order m of the cyclic group is greater than 4, we obtain Theorem 1 using mainly the Riemann-Hurwitz relation which is classical but powerful. In § 2 and § 3 we study Riemann surfaces which are defined by y3= x (xl)(x-tl)(x-t2) and y4= x (xl)(xt). We obtain Theorem 2 and Theorem 3 by considering Weierstrass points of these Riemann surfaces and canonical models of them in~ 2. In § 4 we consider special Riemann surfaces which are characterized by Weierstrass points. We obtain concretely the groups of automorphisms of these Riemann surfaces and the equations by which these surfaces are defined. In § 5 we study Riemann surfaces which are cyclic coverings of complex tori. We obtain Theorem 4 where some new curves come forward. The main result is as follows: