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Takeshi Harui

Bio: Takeshi Harui is an academic researcher from Osaka University. The author has contributed to research in topics: Plane curve & Automorphism. The author has an hindex of 5, co-authored 10 publications receiving 47 citations.

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TL;DR: In this article, the authors classified finite groups acting on smooth plane curves of degree at least four and gave an upper bound for the order of automorphism groups of smooth plane curve and determined the exceptional cases in terms of defining equations.
Abstract: The author classifies finite groups acting on smooth plane curves of degree at least four. Furthermore, he gives an upper bound for the order of automorphism groups of smooth plane curves and determines the exceptional cases in terms of defining equations. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximum order for each degree.

16 citations

Journal ArticleDOI
TL;DR: In this paper, the quotient curves of smooth plane curves with automorphisms are shown to be extremal curves in the sense of Castelnuovo's bound, and they can be divided into two types (type I and type II).
Abstract: We obtain several results of quotient curves of smooth plane curves with automorphisms. Such automorphisms can be divided into two types (type I and type II). The quotient curves of smooth plane curves with automorphisms of type I are extremal curves in the sense of Castelnuovo's bound. We also show some partial result on automorphisms of type II and give examples.

10 citations

Journal ArticleDOI
01 Jun 2018
TL;DR: In this article, the authors constructed typical examples of smooth plane curves C by applying the method of Galois points, whose automorphism group has order 60d, and determined the structure of the automomorphism group of those curves.
Abstract: Recently, the first author [3] classified finite groups obtained as automorphism groups of smooth plane curves of degree d ≥ 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{2d(d – 2), 60d}. 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 60d. In fact, we determine the structure of the automorphism group of those curves.

8 citations

Journal ArticleDOI
Takeshi Harui1
TL;DR: In this paper, it was shown that the gonality of curves on an elliptic ruled surface is twice the degree of the restriction of the bundle map and the Clifford index of such curves is computed by pencils of minimal degree, under certain numerical conditions.
Abstract: In this paper it is shown that the gonality of curves on an elliptic ruled surface is twice the degree of the restriction of the bundle map and the Clifford index of such curves is computed by pencils of minimal degree, under certain numerical conditions. It is also proved that any pencil computing the gonality and the Clifford index of curves is composed with the restriction of the bundle map under some stronger conditions. On the other hand, we found some counterexample to the constancy of gonality and Clifford index in a linear system.

7 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that there exist no double coverings between two smooth plane curves, except for several special cases, and that they are not necessary for any smooth plane curve.
Abstract: We completely classify the pairs of two smooth plane curves with double coverings between them. More precisely, we show that there exist no double coverings between two smooth plane curves except for several special cases.

6 citations


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TL;DR: In this paper, the Torelli theorem for cubic four-folds and the classification of the fixed-point sublattices of the Leech lattice were used to classify the symplectic automorphism groups for cubic 4-fold automorphisms.
Abstract: We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

22 citations

Posted Content
TL;DR: In this paper it was shown that every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting $C$ along a lattice direction.
Abstract: Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $\Delta$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of $\Delta$. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is 'combinatorial', in the sense that it corresponds to projecting $C$ along a lattice direction. We then give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension $2$, and that if a non-hyperelliptic smooth projective curve $C$ of genus $g \geq 2$ can be embedded in the $n$th Hirzebruch surface $\mathcal{H}_n$, then $n$ is actually an invariant of $C$.

22 citations

Journal ArticleDOI
TL;DR: In this article, the authors determined the loci corresponding to non-singular degree projective plane curves, which are non-empty, and presented the analogy of Henn's results for quartic curves concerning nonsingular plane model equations associated to these loci.
Abstract: Henn and Komiya-Kuribayashi listed, independently, the groups $G$ for which $\widetilde{M_3^{Pl}(G)}$ is non-empty. In this paper, we determine the loci $\widetilde{M_6^{Pl}(G)}$, corresponding to non-singular degree $5$ projective plane curves, which are non-empty. Also, we present the analogy of Henn's results for quartic curves concerning non-singular plane model equations associated to these loci.

21 citations

Journal ArticleDOI
TL;DR: In this article it was shown that every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction.
Abstract: Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta. It is classical that the geometric genus of C equals the number of lattice points in the interior of Delta. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index, and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated with a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g >= 2 can be embedded in the nth Hirzebruch surface H-n, then n is actually an invariant of C. This article comes along with three Magma files: basic_commands.m, gonal.m, neargonal.m

21 citations

Journal ArticleDOI
TL;DR: In this article, the loci corresponding to nonsingular degree 5 projective plane curves, which are nonempty, were determined. But these loci do not cover the full automorphism group of genus g curves.
Abstract: Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite nontrivial group) is isomorphic to a subgroup of Aut(δ), and let be the subset of curves δ such that G ≅ Aut(δ), where Aut(δ) is the full automorphism group of δ. Now, for an integer d ≥ 4, let be the subset of Mg representing smooth, genus g, plane curves of degree d, i.e. smooth curves that admits a plane non-singular model of degree d, (in this case, g = (d − 1)(d − 2)/2), and consider the sets and .Henn in [7] and Komiya and Kuribayashi in [10], listed the groups G for which is nonempty. In this article, we determine the loci , corresponding to nonsingular degree 5 projective plane curves, which are nonempty. Also, we present the analogy of Henn's results for quartic curves concerning nonsingular plane model equations associated to these loci (see Table 2 for more details). Similar arguments can be applied to deal with h...

19 citations