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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 $$\lambda\neq0,1$$.

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