Some remarks on bielliptic and trigonal curves

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We prove some results on algebraic curves X of genus g≥ 2 in characteristic 0. For example: Assume that X has an automorphism σ of prime order p≥ 5. If σ has no fixed points, then X cannot be trigonal. On the other hand, if σ has fixed points, then X is bielliptic only if it belongs to one of three extremal types of curves of small genus.