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

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 ${\pi }_P:C⇢{\mathbb{P}}^1$ 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 $C^{\ast }\subset {\mathbb{P}}^{2\ast }$ which preserves the fibers of the projection ${\pi }_{\overline{P}}$ from a certain point $\overline{P}\in {\mathbb{P}}^{2\ast }$. 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 $\mathrm{deg}({\pi }_P)$ is prime and $\mathrm{deg}({\pi }_{\overline{P}})=2\mathrm{deg}({\pi }_P)$. 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.

arXiv comment15 pages

Cited by in corpus (1)

• Quasi-Galois points