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Satoru Fukasawa

Publications and source records attributed to Satoru Fukasawa.

At least 37 records · Page 2Linked to original sources
arXiv2018

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of $q$ elements, which are Frobenius nonclassical for different powers of $q$. This family is an important class of plane curves with many remarkable properties. It contains the Dickson-Guralnick-Zieve curve, which has been recently studied by Giulietti, Korchmaros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.

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arXiv2018

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.

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arXiv2018

Galois lines for the Giulietti--Korchmáros curve

We describe the arrangement of all Galois lines for the Giulietti--Korchmáros curve in the projective $3$-space. As an application, we determine the set of all Galois points for a plane model of the GK curve. This curve possesses many Galois points.

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arXiv2016

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.

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arXiv2016

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.

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arXiv2016

Quasi-Galois points

We introduce the new notion of the "quasi-Galois point" in Algebraic geometry, which is a generalization of the Galois point. A point $P$ in projective plane is said to be quasi-Galois for a plane curve if the curve admits a non-trivial birational transformation which preserves the fibers of the projection $π_P$ from $P$. We discuss the standard form of the defining equation of curves with quasi-Galois points, the number of quasi-Galois points, the structure of the Galois group for the projection, relations with dual curves, and so on. Our theory also has applications to the study of automorphism groups of algebraic curves.

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arXiv2015

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.

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arXiv2015

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.

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arXiv2015

Galois points for a plane curve and its dual curve, II

Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A point $P$ in projective plane is said to be Galois if the function field extension induced by the projection $π_P: C \dashrightarrow \mathbb P^1$ from $P$ is Galois. Further we say that a Galois point is extendable if any birational transformation induced by the Galois group can be extended to a linear transformation of the projective plane. This article is the second part of [2], where we showed that the Galois group at an extendable Galois point $P$ has a natural action on the dual curve $C^* \subset \mathbb{P}^{2*}$ which preserves the fibers of the projection $π_{\overline{P}}$ from a certain point $\overline{P} \in \mathbb{P}^{2*}$. In this article we improve such a result, and we investigate the Galois group of $π_{\overline{P}}$. In particular, we study both when $\overline{P}$ is a Galois point, and when ${\rm deg} \ (π_P)$ is prime and ${\rm deg} \ (π_{\overline{P}}) = 2{\rm deg} \ (π_P)$. As an application, we determine the number of points at which the Galois groups are certain fixed groups for the dual curve of a cubic curve.

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arXiv2014

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.

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arXiv2011

Characterization of a Hermitian curve by Galois point

For a plane curve, a point in the projective plane is said to be Galois when the point projection induces a Galois extension of function fields. We give a new characterization of a Fermat curve whose degree minus one is a power of $p$ in characteristic $p>2$, which is sometimes called Hermitian, by the number of Galois points lying on the curve.

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arXiv2011

Rational curves with many rational points over a finite field

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented.

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