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The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated., 14 pages. Main Theorems are extended

We describe the arrangement of all Galois lines for the Giulietti–Korchmaros 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.

A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described., 7 pages; Changed the title, Added the reference [10], Made some minor changes

We investigate the automorphism group of a plane curve, introducing the notion of a quasi-Galois point. We show that the automorphism group of several curves, for example, Klein quartic, Wiman sextic and Fermat curves, is generated by the groups associated with quasi-Galois points.

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. As an application, it is proved that the Hermitian curve in positive characteristic coincides with the Galois closures of projections of some plane curve from some two non-uniform points., 6 pages

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 $\pi_P: C \dashrightarrow \mathbb P^1$ 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 $\pi_{\overline{P}}$ 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 $\pi_{\overline{P}}$. In particular, we study both when $\overline{P}$ is a Galois point, and when ${\rm deg} \ (\pi_P)$ is prime and ${\rm deg} \ (\pi_{\overline{P}}) = 2{\rm deg} \ (\pi_P)$. 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., Comment: 15 pages

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti-Korchmaros curve or of the curves constructed by Skabelund. They are new examples of plane curves with two Galois points., 9 pages

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., 7 pages

Let C be an irreducible plane curve. A point P in the projective plane is said to be Galois with respect to C if the function field extension induced by the projection from P is Galois. We denote by δ ′ ( C ) the number of Galois points contained in P 2 ∖ C . In this article we will present two results with respect to determination of δ ′ ( C ) in characteristic two. First we determine δ ′ ( C ) for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ ′ ( C ) . Second we determine δ ′ ( C ) for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $\Delta(X)$ is a finite set, and prove that $X$ is a cone if $\Delta(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: For example, we can classify the Galois group induced from a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$., Comment: 19 pages

We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p . In particular, it is shown that any variety in p > 0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p = 2 .

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(pe + 1) of degree pe + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \({\mathbb{F}_{2}}\) -rational points in \({\mathbb{P}^{2}}\).

In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙN to re-embed into some projective space ℙM so as to be non-reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙN is identically zero; hence the projective varietyX re-embedded in ℙM yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙN with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙN satisfying the assumption. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study the reflexivity of a Segre product of a projective space \({\mathbb P^m}\) and a projective variety Y, and give a criterion for \({\mathbb P^m \times Y}\) to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.

For a smooth plane curve $$C \subset {\mathbf{P}}^2$$ , we call a point $$P \in {\mathbf{P}}^2$$ a Galois point if the point projection $$\pi_P:C \rightarrow {\mathbf{P}}^1$$ at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.

We find a nonreflexive projective 3-fold in P 5 with birational Gauss map onto its image, in any positive characteristic. This gives a negative answer, also in characteristic p > 2, of the following Kleiman–Piene's question: Does the separability of the Gauss map imply reflexivity? Kaji already gave a negative answer in characteristic 2.

It is known that if a projective variety X in PN is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

For any given projective variety Y, we construct a projective variety Open image in new window whose general fiber of the Gauss map with reduced scheme structure is isomorphic to Y when the characteristic >0.

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., 7 pages

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 completely classify plane curves with infinitely many outer Galois points.

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 pages

We find a characteristic-free algebraic condition for developability of uniruled varieties. As an application, we study developable varieties over positive characteristic fields. In particular, we generalize classification theorem of one parameter developable ruled varieties to arbitrary characteristic.

For a plane curve C, we call a point P∈P2 a Galois point with respect to C if the point projection from P induces a Galois extension of function fields. We give an example of a singular plane curve having infinitely many inner and outer Galois points. We also classify plane curves whose general points are inner Galois points. Before our results, known examples in the theory of Galois points have only finitely many Galois points, except trivial cases.