Galois points on quartic surfaces. (English) Zbl 1067.14510

Let \(S\) be a smooth hypersurface in the projective three space and consider a projection of \(S\) from a point \(P\in S\) to a plane \(H\). This projection induces a field extension \(k(S)/k(H)\). \(P\) is called a Galois point if this extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is 0,1,2,4 or 8 and the existence of some rule of distribution of the Galois points. Especially there exists only one surface with eight Galois points, which is defined by \(xy^3+zw^3+x^4+z^4=0\) and has 64 lines.


14J28 \(K3\) surfaces and Enriques surfaces
12F10 Separable extensions, Galois theory
14J70 Hypersurfaces and algebraic geometry
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