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

TL;DR

This paper investigates Galois points on plane curves and their duals, focusing on the Galois groups' actions, extending previous results, and applying findings to cubic curves to determine fixed Galois group points.

Contribution

It improves understanding of Galois groups at dual points and explores conditions when these points are Galois, especially for prime degree projections and dual curves.

Findings

Galois group actions on dual curves are characterized.

Conditions for dual points to be Galois points are established.

Number of fixed Galois group points on dual cubic curves is determined.

Abstract

Let $C \subset \mathbb{P}^2$ 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 $\pi_P: C \dashrightarrow \mathbb P^1$ 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 $C^* \subset \mathbb{P}^{2*}$ which preserves the fibers of the projection $\pi_{\overline{P}}$ from a certain point $\overline{P} \in \mathbb{P}^{2*}$. In this article we improve such a result, and we investigate the Galois group of $\pi_{\overline{P}}$. In particular, we study both when $\overline{P}$ is a Galois point, and when ${\rm deg} \ (\pi_P)$ is prime and ${\rm deg} \ (\pi_{\overline{P}}) = 2{\rm deg} \ (\pi_P)$. 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.