Birational embeddings of the Hermitian, Suzuki and Ree curves with two Galois points

We show that there exists a plane curve of degree $q^3+1$ with two inner Galois points whose smooth model is the Hermitian curve of degree $q+1$, where $q$ is a power of the characteristic $p>0$. 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

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

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

A hyperplane section theorem for Galois points and its application

A point $P$ in projective space is said to be Galois with respect to a hypersurface if the function field extension induced by the projection from $P$ is Galois. We present a hyperplane section theorem for Galois points. Precisely, if $P$ is a Galois point for a hypersurface, then $P$ is Galois for a general hyperplane section passing through $P$. As an application, we determine hypersurfaces of dimension $n$ with $n$-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 $C$ 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 $\delta(C)$ (resp. $\delta'(C)$) the number of Galois points contained in $C$ (resp. in $\mathbb P^2 \setminus C$). In this article, we determine the numbers $\delta(C)$ and $\delta'(C)$ in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth plane curves by the number $\delta(C)$ or $\delta'(C)$. In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number $\delta'(C)$.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