# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 129, 2013, pp. 93–113**

**DOI: 10.4171/RSMUP/129-7**

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

Satoru Fukasawa^{[1]}(1) Yamagata University, Japan

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. ${\delta} '(C)$) 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 ${\delta} '(C)$ 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 ${\delta} '(C)$. In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number ${\delta} '(C)$.

*Keywords: *Galois point, plane curve, positive characteristic, Galois group

Fukasawa Satoru: Complete Determination of the Number of Galois Points for a Smooth Plane Curve. *Rend. Sem. Mat. Univ. Padova* 129 (2013), 93-113. doi: 10.4171/RSMUP/129-7