Some remarks on bielliptic and trigonal curves

TL;DR

The paper investigates properties of algebraic curves of genus at least 2 over characteristic zero, focusing on automorphisms of prime order and their implications for the curves being trigonal or bielliptic.

Contribution

It establishes new restrictions on the structure of algebraic curves with automorphisms of prime order, linking fixed points and automorphism order to trigonal and bielliptic properties.

Findings

Curves with prime order automorphisms without fixed points are not trigonal.

Curves with fixed points and prime order automorphisms are bielliptic only in specific small genus cases.

Automorphism properties impose strong geometric constraints on algebraic curves.

Abstract

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.