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

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 $π_P: C \dashrightarrow \mathbb P^1$ 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 $π_{\overline{P}}$ 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 $π_{\overline{P}}$. In particular, we study both when $\overline{P}$ is a Galois point, and when ${\rm deg} \ (π_P)$ is prime and ${\rm deg} \ (π_{\overline{P}}) = 2{\rm deg} \ (π_P)$. 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.