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Galois points for a plane curve in characteristic two. (English) Zbl 1287.14015
Let \(K\) be an algebraically closed field of characteristic \(p>0\) and let \(C\subset \mathbb{P}^2_K\) be a plane curve of degree \(d\geq4\). A point \(P\in\mathbb{P}^2_K\) is said to be Galois if the function field extension \(K(C)/K(\mathbb{P}^1)\) determined by the projection \(C\to\mathbb{P}^1\) from \(P\) is a Galois extension. Denote by \(\delta'(C)\) the number of outer Galois points; that is, those not belonging to \(C\).
In this paper, this invariant \(\delta'(C)\) is determined for smooth curves in the case \(p=2\), \(d=2^e\). The author proves that \(\delta'(C)=0,1,3\) or \(7\). The case \(\delta'(C)=7\) occurs only when \(d=4\) and \(C\) is projectively equivalent to the Klein quartic. The case \(\delta'(C)=3\) occurs only when \(d=4\) and \(C\) is projectively equivalent to the curve \((x^2+x)^2+(x^2+x)(y^2+y)+(y^2+y)^2=\lambda\), for \(\lambda\neq0,1\).

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