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Function field theory of plane curves by dual curves. (English) Zbl 1064.14023
Summary: We study the structure of function fields of plane curves following our method developed previously in a joint paper with K. Miura [J. Algebra 226, 283–294 (2000; Zbl 0983.11067)]. Let \(K\) be the function field of a smooth plane curve \(C\) of degree \(d\geq 4\) and let \(K_P\) be a maximal rational subfield of \(K\) for \(P\in \mathbb{P}^2\). We study the field extension \(K/K_P\) from a geometrical viewpoint. Especially, we give a sufficient condition that the Galois group of the Galois closure of \(K/K_P\) becomes a full symmetric group.

MSC:
14H05 Algebraic functions and function fields in algebraic geometry
12F10 Separable extensions, Galois theory
14H50 Plane and space curves
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[1] Iitaka, S., Algebraic geometry, Graduate texts in math., 76, (1982), Springer-Verlag New York/Heidelberg/Berlin
[2] Lang, S., Introduction to algebraic and abelian functions, Graduate texts in math., 89, (1995), Springer-Verlag New York/Heidelberg/Berlin
[3] Miura, K.; Yoshihara, H., Field theory for function fields of plane quartic curves, J. algebra, 226, 283-294, (2000) · Zbl 0983.11067
[4] Miura, K.; Yoshihara, H., Field theory for the function field of the quintic Fermat curve, Comm. algebra, 28, 1979-1988, (2000) · Zbl 0978.14024
[5] Namba, M., Geometry of projective algebraic curves, (1984), Dekker New York/Basel · Zbl 0556.14012
[6] Namba, M., Branched coverings and algebraic functions, Research notes in math., 161, (1987), Pitman-Longman London
[7] M. Oka, Geometry of cuspidal sextics and their dual curves, to appear. · Zbl 1020.14008
[8] Serre, J.P., Notes written by H. Darmon, “topics in Galois theory”, Research notes in math., 1, (1992), Jones and Bartlett Boston/London · Zbl 0746.12001
[9] Tokunaga, H., Triple coverings of algebraic surfaces according to the Cardano formula, J. math. Kyoto univ., 31, 359-375, (1991) · Zbl 0754.14010
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