60 research outputs found
Birational embeddings of the Hermitian, Suzuki and Ree curves with two Galois points
We show that there exists a plane curve of degree with two inner
Galois points whose smooth model is the Hermitian curve of degree , where
is a power of the characteristic . Similar results hold for the Suzuki
and Ree curves respectively.Comment: 7 page
A family of plane curves with two or more Galois points in positive characteristic
We give new examples of plane curves with two or more Galois points as a
family, and describe the number of Galois points for these curves, by using
finite fields.Comment: 13 pages. Changed the title and Revised Lemmas 4, 5 and
On the number of Galois points for a plane curve in characteristic zero
For a plane curve, a point on the projective plane is said to be Galois if
the projection from the point as a map from the curve to a line induces a
Galois extension of function fields. We present upper bounds for the number of
Galois points, if the genus is greater than zero. If the curve is not an
immersed curve, then we have at most two Galois points. If the degree is not
divisible by two nor three, then the number of outer Galois points is at most
three. As a consequence, a conjecture of Yoshihara is true in these cases.Comment: 7 page
Rational points and Galois points for a plane curve over a finite field
We study the relationship between rational points and Galois points for a
plane curve over a finite field. It is known that the set of Galois points
coincides with that of rational points of the projective plane if the curve is
the Hermitian, Klein quartic or Ballico-Hefez curves. We propose a problem:
Does the converse hold true? When the curve of genus at most one has a rational
point, we will have an affirmative answer.Comment: 7 page
Bounds for the number of Galois points for plane curves
A point on a plane curve is said to be Galois (for the curve) if the
projection from the point as a map from the curve to a line induces a Galois
extension of function fields. It is known that the number of Galois points is
finite except for a certain explicit example. We establish upper bounds for the
number of Galois points for all plane curves other than the example in terms of
the genus, degree and the generic order of contact, and settle curves attaining
the bounds.Comment: 16 pages; Extended Main Theorem and added two corollarie
Galois points for the Dickson-Guralnick-Zieve curve
The Dickson-Guralnick-Zieve curve over a finite field has been studied
recently by Giulietti, Korchmaros and Timpanella in several points of view. In
this short note, the distribution of Galois points for this curve is
determined. As a consequence, a problem posed by the present author in the
theory of Galois point is modified.Comment: 3 pages, Fixed typos in Fact
A hyperplane section theorem for Galois points and its application
A point in projective space is said to be Galois with respect to a
hypersurface if the function field extension induced by the projection from
is Galois. We present a hyperplane section theorem for Galois points.
Precisely, if is a Galois point for a hypersurface, then is Galois for
a general hyperplane section passing through . As an application, we
determine hypersurfaces of dimension with -dimensional sets of Galois
points.Comment: 10 pages, changed the titl
Complete determination of the number of Galois points for a smooth plane curve
Let be a smooth plane curve. A point in the projective plane is said
to be Galois with respect to if the function field extension induced from
the point projection from is Galois. We denote by (resp.
) the number of Galois points contained in (resp. in ). In this article, we determine the numbers and
in any remaining open cases. Summarizing results obtained by now,
we will have a complete classification theorem of smooth plane curves by the
number or . In particular, we give new
characterizations of Fermat curve and Klein quartic curve by the number
.Comment: 16 page
Rational curves of degree four with two inner Galois points
We characterize plane rational curves of degree four with two or more inner
Galois points. A computer verifies the existence of plane rational curves of
degree four with three inner Galois points. This would be the first example of
a curve with exactly three them. Our result implies that Miura's bound is sharp
for rational curves.Comment: 8 page
Automorphism groups of smooth plane curves with many Galois points
We settle the automorphism groups of curves appearing in a classification
list of smooth plane curves with at least two Galois points. One of them is an
ordinary curve whose automorphism group exceeds the Hurwitz bound.Comment: 6 page
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