# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 132, 2014, pp. 61–74**

**DOI: 10.4171/RSMUP/132-5**

Published online: 2014-11-04

Galois points for a plane curve and its dual curve

Satoru Fukasawa^{[1]}and Kei Miura

^{[2]}(1) Yamagata University, Japan

(2) Ube National College of Technology, Ube, Yamaguchi, Japan

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: {\it 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^eZ^{d-e}=0$ , which is a famous self-dual curve.

*Keywords: *Galois point, plane curve, dual curve, self-dual curve

Fukasawa Satoru, Miura Kei: Galois points for a plane curve and its dual curve. *Rend. Sem. Mat. Univ. Padova* 132 (2014), 61-74. doi: 10.4171/RSMUP/132-5