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Die Automorphismengruppen hyperelliptischer Kurven. (The groups of automorphisms of hyperelliptic curves). (German) Zbl 0588.14022
Let \(\Gamma\) be a hyperelliptic curve over an algebraically closed field of characteristic 0 and G a group of automorphisms of \(\Gamma\) containing the canonical involution. The authors study the group theoretic structure of G and relate it to the ramification behavior of the morphism \(p:\quad \Gamma \to \Gamma /G.\) Using the well known classification of finite subgroups of PGL(2,\({\mathbb{C}})\), they first obtain a classification of the possible ramification types of p. The computation of the number of involutions in G then shows the main result (theorem 5.1): Up to isomorphism, G depends only on the ramification type of p.
Actually, the authors do much more: For each type, the corresponding group is determined, and a way to obtain normal forms is indicated. The proofs use complicated case by case considerations which are carried out only in typical special cases.
Reviewer: E.-U.Gekeler

MSC:
14H45 Special algebraic curves and curves of low genus
14L30 Group actions on varieties or schemes (quotients)
14H30 Coverings of curves, fundamental group
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References:
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