Automorphism group of plane curve computed by Galois points

Abstract

Let $C \subset {\mathbb P}^2$ be a smooth plane curve, and $P_1, \ldots , P_m$ be all inner and outer Galois points for $C$. Each Galois point $P_i$ determines a Galois group at $P_i$, say $G_{P_i}$. Then, by the definition of Galois point, an element of the Galois group $G_{P_i}$ induces a birational transformation of $C$. In fact, we see that it becomes an automorphism of $C$. We call this an automorphism belonging to the Galois point $P_i$. Then, we consider the group $G(C)$ generated by automorphisms belonging to all Galois points for $C$. In particular, we investigate the difference between $\mathrm{Aut} (C)$ and $G(C)$, so that we determine the structure of $\mathrm{Aut} (C)$.