Some remarks on bielliptic and trigonal curves

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