## 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.

### MSC:

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