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A note on Galois embedding and its application to \(\mathbb{P}^n\). (English) Zbl 1391.14107

Summary: We show a condition that a Galois covering \(\pi:V\to\mathbb{P}^n\) is induced by a Galois embedding. Then we consider the Galois embedding for \(\mathbb{P}^n\). If the Galois group \(G\) is abelian, then \(G\cong\bigoplus\limits^{n}Z_d\) and the projection \(\pi\) can be expressed as \(\pi(X_0:X_1:\cdots:X_n)=(X_0^d:X_1^d:\cdots:X_n^d)\).

MSC:

14N05 Projective techniques in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14J99 Surfaces and higher-dimensional varieties

References:

[1] I. Dolgachev, Reflection group in algebraic geometry, Bull. Amer. Math. Soc. 45 (2008), 1-60. · Zbl 1278.14001 · doi:10.1090/S0273-0979-07-01190-1
[2] J. Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), 685-724. · Zbl 0433.14040
[3] M. Namba, Branched coverings and algebraic functions, Pitman Research Notes in Mathematics Series 161. · Zbl 0706.14017
[4] H. Yoshihara, Galois points for plane rational curves, Far East J. Math. Sci. 25 (2007), 273-284. · Zbl 1132.14309
[5] H. Yoshihara, Galois embedding of algebraic variety and its application to abelian surface, Rend. Semin. Mat. Univ. Padova 117 (2007), 69-85. · Zbl 1136.14041
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