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Complete determination of the number of Galois points for a smooth plane curve. (English) Zbl 1273.14066
Let \(C \subset \mathbb{P}^2\) be a smooth plane curve of degree \(d \geq 4\) over an algebraically closed base field. Let \(P \in \mathbb{P}^2\) be a point. Projection from \(P\) onto a line \(\mathbb{P}^1 \subset \mathbb{P}^2\) defines a field extension \(K(C) / K(\mathbb{P}^1)\). \(P\) is called a Galois point with respect to \(C\) if this field extension is Galois. If \(P \in C\), \(P\) is called an inner Galois point; otherwise, it is called an outer Galois point. The number of inner (outer) Galois points is denoted \(\delta(C)\) (\(\delta^\prime(C)\)). These numbers have been the subject of investigation in a number of papers by Yoshihara, Miura, Homma and the author of the present paper. (A list of references can be found in the introduction.) In the paper under review, the author completes the determination of \(\delta(C)\) and \(\delta^\prime(C)\). The results are summarized in Theorem 3. (It must, however, be mentioned that no distinction is stated between \(\delta(C)=0\) and \(\delta(C)=1\); and similarly between \(\delta^\prime(C)=0\) and \(\delta^\prime(C)=1\).) The methods used in the paper are elementary.

MSC:
14H50 Plane and space curves
12F10 Separable extensions, Galois theory
14H05 Algebraic functions and function fields in algebraic geometry
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