The author classifies finite groups acting on smooth plane curves of degree at least four. Furthermore, he gives some upper bounds for the order of automorphism groups of smooth plane curves and determines the exceptional cases in terms of defining equations. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximum order for each degree.

No abstract

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g = 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.

Abstract. Under certain numerical conditions, the gonality of curves on an elliptic ruled surface is twice the degree of the bundle map of the ruled surface and the Clifford index of such curves is computed by pencils of minimal degree. IntroductionThroughout this paper, a curve or a surface always means a projective, reduced and irreducible one over C. This is a non-negative integer by Clifford's Theorem. A divisor D is said to compute the Clifford index of X if D contributes to the Clifford index of X and satisfies that

Recently, the first author [3] classified finite groups obtained as automorphism groups of smooth plane curves of degree d ≥ 4 into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by max{2d(d − 2), 60d}. In this article, we shall construct typical examples of smooth plane curve C by applying the method of Galois points, whose automorphism group has order 60d. In fact, we determine the structure of the automorphism group of those curves.