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 $$\mathrm{Aut} (C)$$ .