- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:49
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
132
2014
0
Galois points for a plane curve and its dual curve
Satoru
Fukasawa
Yamagata University, YAMAGATA, JAPAN
Kei
Miura
Ube National College of Technology, UBE, YAMAGUCHI, JAPAN
Galois point, plane curve, dual curve, self-dual curve
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.
Algebraic geometry
Field theory and polynomials
61
74
10.4171/RSMUP/132-5
http://www.ems-ph.org/doi/10.4171/RSMUP/132-5