Complete Determination of the Number of Galois Points for a Smooth Plane Curve

Abstract

Let $C$ be a smooth plane curve. A point $P$ in the projective plane is said to be Galois with respect to $C$ if the function field extension induced by the projection from $P$ is Galois. We denote by $\delta (C)$ (resp. ${\delta }^{\prime }(C)$) the number of Galois points contained in $C$ (resp. in ${\mathbb{P}}^2\setminus C$). In this article, we determine the numbers $\delta (C)$ and ${\delta }^{\prime }(C)$ in any remaining open cases. Summarizing results obtained by now, we will present a complete classification theorem of smooth plane curves by the number $\delta (C)$ or ${\delta }^{\prime }(C)$. In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number ${\delta }^{\prime }(C)$.