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In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.

For a plane curve C , we call a point P ∈ P 2 a Galois point with respect to C if the point projection from P induces a Galois extension of function fields. We give an example of a singular plane curve having infinitely many inner and outer Galois points. We also classify plane curves whose general points are inner Galois points. Before our results, known examples in the theory of Galois points have only finitely many Galois points, except trivial cases.

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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 from the point projection from P is Galois. We denote by δ(C) (resp. δ ′ (C)) the number of Galois points contained in C (resp. in P 2 \ C). In this article, we determine the numbers δ(C) and δ ′ (C) in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth plane curves by the number δ(C) or δ ′ (C). In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number δ ′ (C). IntroductionLet the base field K be an algebraically closed field of characteristic p ≥ 0 and let C ⊂ P 2 be a smooth plane curve of degree d ≥ 4. In 1996, H. Yoshihara introduced the notion of Galois point (see [14,18] or survey paper [5]). If the function field extension K(C)/K(P 1 ), induced from the projection π P : C → P 1 from a point P ∈ P 2 , is Galois, then the point P is said to be Galois with respect to C. When a Galois point P is contained in C (resp. P 2 \ C), we call P an inner (resp. outer) Galois point. We denote by δ(C) (resp. δ ′ (C)) the number of inner (resp. outer) Galois points for C. It is remarkable that many classification results of algebraic varieties have been given in the theory of Galois point.Yoshihara determined δ(C) and δ ′ (C) in characteristic p = 0 ([14, 18] Problem.(1) Let p = 2 and let e ≥ 2. Find and classify smooth plane curves of degree(2) Let p > 0, e ≥ 1 and let d = p e l, where l is not divisible by p. Assume that (p e , l) = (p, 1), (2 e , 1). Then, determine δ ′ (C).In this article, we give a complete answer to these problems. where c ∈ K \ {0, 1}.Theorem 2. Let the characteristic p > 0, let e ≥ 1, let l be not divisible by p, and let C be aSummarizing Theorems 1 and 2 and the results of Yoshihara, Homma and the present author, we will have the following classification theorem of smooth plane curves by the number δ(C) orTheorem 3 (Yoshihara, Homma, Fukasawa). Let C be a smooth plane curve of degree d ≥ 4 in characteristic p ≥ 0. Then:Furthermore, we have the following. 2 . Furthermore, we have the following. This is a modified and extended version of the paper [4, Part IV] (which will have been published only in arXiv). PreliminariesLet C ⊂ P 2 be a smooth plane curve of degree d ≥ 4 in characteristic p > 0. For a point P ∈ C,we denote by T P C ⊂ P 2 the (projective) tangent line at P . For a projective line l ⊂ P 2 and a COMPLETE DETERMINATION OF THE NUMBER OF GALOIS POINTS FOR A SMOOTH PLANE CURVE 3point P ∈ C ∩ l, we denote by I P (C, l) the intersection multiplicity of C and l at P . We denote by P R the line passing through points P and R when P = R, and by π P : C → P 1 ; R → P R the point projection from a point P ∈ P 2 . If R ∈ C, we denote by e R the ramification index of π P at R. It is not difficult to check the following.Lemma 1. Let P ∈ P 2 and let R ∈ C. Then for π P we have the following.(1) If R = P , then e R = I R (C, T R C) − 1.(2)...