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

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• 作者：Satoru Fukasawa^a ; ¹ ; ^{s.fukasawa@sci.kj.yamagata-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Kei Miura^b ; ² ; ^{kmiura@ube-k.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14H50 ; 14H05 ; 12F10
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：220
• 期：5
• 页码：2038-2048
• 全文大小：366 K

文摘

Let C⊂P²$C\subset {\mathbb{P}}^2$ 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⇢P¹${\pi }_P:C⇢{\mathbb{P}}^1$ 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^⁎⊂P^2⁎$C^{⁎}\subset {\mathbb{P}}^{2⁎}$ which preserves the fibers of the projection ${\pi }_{\bar{P}}$ from a certain point $\bar{P}\in {\mathbb{P}}^{2⁎}$. In this article we improve this result, and we investigate the Galois group of ${\pi }_{\bar{P}}$. In particular, we study both when $\bar{P}$ is a Galois point, and when deg⁡(π_P)$\mathrm{deg}({\pi }_P)$ is prime and $\mathrm{deg}({\pi }_{\bar{P}})=2\mathrm{deg}({\pi }_P)$. 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.