arXiv subjects
Publications and source records attributed to Kei Miura.
We study quasi-Galois points, which are a generalization of Galois points. We characterize smooth quartic surfaces admitting a quasi-Galois point in terms of K3 surfaces with a certain involution, and provide a criterion for quasi-Galois points for quartic surfaces. Moreover, we also determine all quasi-Galois points of the Fermat quartic surface and show that it has exactly 28 quasi-Galois points.
We prove that there exists a one-to-one correspondence between smooth quartic surfaces with an outer Galois point and K3 surfaces with a certain automorphism of order 4. Furthermore, we characterize quartic surfaces with two or more outer Galois points as K3 surfaces.
We prove that there exists a one to one correspondence between smooth quartic surfaces with an inner Galois point and Eisenstein $K3$ surfaces of type $(4, 3)$. Furthermore we characterize the quartic surface with 8 (the maximum number) inner Galois points as a singular $K3$ surface.
We study birational transformations belonging to Galois points. Let $P$ be a Galois point for a plane curve $C$ and $G_P$ be a Galois group at $P$. Then an element of $G_P$ induces a birational transformation of $C$. In general, it is difficult to determine when this birational transformations can be extended to a Cremona (or projective) transformation. In this article, we shall prove that if the Galois group is isomorphic to the cyclic group of order three, then any element of the Galois group has an expression as a de Jonquières transformation. In particular, they can be extended to Cremona transformations.
In Part I, the present authors introduced the notion of a quasi-Galois point, for investigating the automorphism groups of plane curves. In this second part, the number of quasi-Galois points for smooth plane curves is described. In particular, sextic or quartic curves with many quasi-Galois points are characterized.
In the quest for quantum spin liquids, thin films are expected to open the way for the control of intricate magnetic interactions in actual materials by exploiting epitaxial strain and two-dimensionality. However, materials compatible with conventional thin-film growth methods have largely remained undeveloped. As a promising candidate towards the materialization of quantum spin liquids in thin films, we here present a robust ilmenite-type oxide with a honeycomb lattice of edge-sharing IrO$_6$ octahedra artificially stabilized by superlattice formation with an ilmenite-type antiferromagnetic oxide MnTiO$_3$. The stabilized sub-unit-cell-thick Mn-Ir-O layer is isostructural to MnTiO$_3$, having the atomic arrangement corresponding to ilmenite-type MnTiO$_3$ not discovered yet. By spin Hall magnetoresistance measurements, we found that antiferromagnetic ordering in the ilmenite Mn sublattice is suppressed by modified magnetic interactions in the MnO$_6$ planes via the IrO$_6$ planes. These findings lay the foundation for the creation of two-dimensional Kitaev candidate materials, accelerating the discovery of exotic physics and applications specific to quantum spin liquids.
In this article, we study birational transformations belonging to Galois points for certain plane quartic curve. In fact, we see that they can be extended to Cremona transformations. In particular, we determine the conjugacy class of them. We have that they are all conjugate to linear transformations.
Recently, the first author classified finite groups obtained as automorphism groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by $\max \left\{ 2 d (d - 2), 60 d \right\}$. In this article, we shall construct typical examples of smooth plane curve $C$ by applying the method of Galois points, whose automorphism group has order $60 d$. In fact, we determine the structure of the automorphism group of those curves.
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.
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.