Galois Point Research Articles

Smooth plane curves with outer Galois points whose reduced automorphism group is $A_{5}$

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group.

LINEAR AUTOMORPHISMS OF SMOOTH HYPERSURFACES GIVING GALOIS POINTS

Let $X$ be a smooth hypersurface $X$ of degree $d\geq4$ in a projective space $\mathbb P^{n+1}$. We consider a projection of $X$ from $p\in\mathbb P^{n+1}$ to a plane $H\cong\mathbb P^n$. This projection induces an extension of function fields $\mathbb C(X)/\mathbb C(\mathbb P^n)$. The point $p$ is called a Galois point if the extension is Galois. In this paper, we will give a necessary and sufficient conditions for $X$ to have Galois points by using linear automorphisms.

Curves with more than one inner Galois point

Let C be an irreducible plane curve of PG(2,K) where K is an algebraically closed field of characteristic p≥0. A point Q∈C is an inner Galois point for C if the projection πQ from Q is Galois. Assume that C has two different inner Galois points Q1 and Q2, both simple. Let G1 and G2 be the respective Galois groups. Under the assumption that Gi fixes Qi, for i=1,2, we provide a complete classification of G=〈G1,G2〉 and we exhibit a curve for each such G. Our proof relies on deeper results from group theory.

Galois points for a plane curve and its dual curve, II

Let C⊂P2 be a plane curve of degree at least three. A point P in projective plane is said to be Galois if the function field extension induced by the projection πP:C⇢P1 from P is Galois. Further we say that a Galois point is extendable if any birational transformation induced by the Galois group can be extended to a linear transformation of the projective plane. This article is the second part of [2], where we showed that the Galois group at an extendable Galois point P has a natural action on the dual curve C⁎⊂P2⁎ which preserves the fibers of the projection πP‾ from a certain point P‾∈P2⁎. In this article we improve this result, and we investigate the Galois group of πP‾. In particular, we study both when P‾ is a Galois point, and when deg⁡(πP) is prime and deg⁡(πP‾)=2deg⁡(πP). As an application, we determine the number of points at which the Galois groups are certain fixed groups for the dual curve of a cubic curve.

Galois points for a plane curve and its dual curve

A point P in projective plane is said to be Galois for a plane curve of degree at least three if the function field extension induced by the projection from P is Galois. Further we say that a Galois point is extendable if any birational transformation by the Galois group can be extended to a linear transformation of the projective plane. In this article, we propose the following problem: If a plane curve has a Galois point and its dual curve has one, what is the curve? We give an answer. We show that the dual curve of a smooth plane curve does not have a Galois point. On the other hand, we settle the case where both a plane curve and its dual curve have extendable Galois points. Such a curve must be defined by X^d-Y^eZ^{d-e}=0 , which is a famous self-dual curve.

Automorphism group of plane curve computed by Galois points

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

Galois points for a normal hypersurface

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $\Delta(X)$ is a finite set, and prove that $X$ is a cone if $\Delta(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: For example, we can classify the Galois group induced from a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.

Galois Lines for Normal Elliptic Space Curves, II

For each linearly normal elliptic curve C ⊂ ℙ3, we determine Galois lines and their arrangement. We prove that the curve C has exactly six V4-lines. In case j(C) = 1, it has eight Z4-lines in addition. The V4-lines form the edges of a tetrahedron. In case j(C) = 1, for each vertex of the tetrahedron, there exist exactly two Z4-lines passing through it. As a corollary we obtain that each plane quartic curve of genus 1 does not have more than one Galois point.

A note on birational transformations belonging to Galois points

Let \({P \in {\mathbb{P}^{n+1}}}\) be a Galois point for a hypersurface \({X \subset {\mathbb{P}}^{n+1}}\) . Then every element of the Galois group GP induces a birational transformation of X. In this note, we study this birational transformation. In particular, for smooth cubic hypersurface, we determine the condition when the birational transformation becomes a linear one.

Rational cuspidal curve with a Galois point

AbstractFor a rational cuspidal curve C we study if it has a Galois point. The result is as follows: if C has an outer Galois point, then C is projectively equivalent to the curve defined by xe = yn where (e, n) = 1. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Singular plane curves with infinitely many Galois points

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.

Rational curve with Galois point and extendable Galois automorphism

Let G be the Galois group of a Galois point for a plane curve C. An element of G induces a birational transformation of C. We study if it can be extended to a projective or birational transformation of the plane. In the course of the study we give the defining equation of a rational curve with the Galois point. Furthermore, we introduce a special birational transformation in order to make the defining equation into a simpler form.

Galois points for a plane curve in arbitrary characteristic

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.

Galois points for plane curves and Cremona transformations

We study the relation between Galois points for plane curves and Cremona transformations. First, we prove that a Galois point is preserved by a standard quadratic transformation. Then, we define a certain minimality for a pair of a plane curve and a Galois point. So that, we study defining equations of plane curves with a Galois point.

On the number of Galois points for a plane curve in positive characteristic, II

For a smooth plane curve $$C \subset {\mathbf{P}}^2$$ , we call a point $$P \in {\mathbf{P}}^2$$ a Galois point if the point projection $$\pi_P:C \rightarrow {\mathbf{P}}^1$$ at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.

Galois points on singular plane quartic curves

We study the structure of function fields of plane curves following our method developed in [K. Miura, H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000) 283–294]. Especially, we study Galois points on singular plane quartic curves and determine the number of Galois points on them. Furthermore, we give concrete defining equations when the curve has a Galois point.

Galois points for smooth hypersurfaces

Let V be a smooth hypersurface in P n+1 . We consider a projection of V from P∈ P n+1 to a hyperplane H. This projection induces an extension of fields k( V)/ k( H), which does not depend on the choice of H. We study the structures of this extension and the hypersurfaces together. The point P is called a Galois point if the extension is Galois. We show estimates of the number of the Galois points and some rules of their distributions. Especially we give the defining equation of V with maximal number of Galois points.

Galois points on quartic surfaces

Let S be a smooth hypersurface in the projective three space and consider a projection of S from P∈S to a plane H. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.

Galois points on quartic surfaces

Let S be a smooth hypersurface in the projective three space and consider a projection of S from P∈S to a plane H. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.