2013
DOI: 10.4171/rsmup/129-7
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Complete Determination of the Number of Galois Points for a Smooth Plane Curve

Abstract: Abstract. Let C be a smooth plane curve. A point P in the projective plane is said to be Galois with respect to C if the function field extension induced from the point projection from P is Galois. We denote by δ(C) (resp. δ ′ (C)) the number of Galois points contained in C (resp. in P 2 \ C). In this article, we determine the numbers δ(C) and δ ′ (C) in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth plane curves by the number δ(C) or δ ′… Show more

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Cited by 18 publications
(11 citation statements)
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“…In p > 0, for the Fermat curve H of degree p e + 1, M. Homma [12] proved that δ(H) = (p e ) 3 + 1. Recently, the present author [3] showed that δ(C) = 0, 1 or d for any other smooth curve C. As a next step, it would be nice to give an upper bound for δ(C) for all irreducible plane curves C. Miura [16] gave a certain inequality related to δ(C) if p = 0 and d − 1 is prime. The present author and T. Hasegawa [6] settled the case δ(C) = ∞.…”
Section: Introductionmentioning
confidence: 92%
“…In p > 0, for the Fermat curve H of degree p e + 1, M. Homma [12] proved that δ(H) = (p e ) 3 + 1. Recently, the present author [3] showed that δ(C) = 0, 1 or d for any other smooth curve C. As a next step, it would be nice to give an upper bound for δ(C) for all irreducible plane curves C. Miura [16] gave a certain inequality related to δ(C) if p = 0 and d − 1 is prime. The present author and T. Hasegawa [6] settled the case δ(C) = ∞.…”
Section: Introductionmentioning
confidence: 92%
“…(1) X I : x q+1 0 + x q+1 1 + x q+1 2 = 0, (2) a nodal curve whose defining equation is given in [4] and [7], (3) strange curves.…”
Section: The Case Of Plane Curvesmentioning
confidence: 99%
“…We say Q ∈ P n+1 is a Galois point if the field extension associated with the projection π Q | X is a Galois extension. Galois points are a particular case of non-uniform points, and have been extensively studied in [8,9,21]. In particular, Theorem 1.1 and Theorem 1.2 of [10] show that Conjecture 1.2 holds for Galois points when X is a normal hypersurface.…”
Section: Introductionmentioning
confidence: 99%