Galois points for a plane curve and its dual curve

Abstract

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 $X^d-Y^e{Z}^{d-e}=0$ , which is a famous self-dual curve.