Author
Takao Kato
Other affiliations: University of Erlangen-Nuremberg
Bio: Takao Kato is an academic researcher from Yamaguchi University. The author has contributed to research in topics: Genus (mathematics) & Plane curve. The author has an hindex of 8, co-authored 33 publications receiving 191 citations. Previous affiliations of Takao Kato include University of Erlangen-Nuremberg.
Papers
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TL;DR: In this paper, the authors generalized Namba's result to the case δ>0 and showed that ifd ≥ 2k+3 and δ
Abstract: LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg
−2/1
and that everyg
−1/1
is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ 0, thenC has no linear seriesg
−3/1
. We also show that ifd≥2k+3 and δ
49 citations
TL;DR: This work presents a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps.
Abstract: In this work we present a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps. We also review results on applications to the theory of error correcting codes. We then recall a generalization of the concept of Weierstrass semigroup, which is the Weierstrass set associated to a linear system and several points. We finish by presenting new results on this Weierstrass set, including some on the cardinality of its set of gaps.
24 citations
TL;DR: In this paper, necessary and sufficient conditions for non-special line bundles of degree2g − 2 and 2g − 3 being not normally generated are given, as well as criteria for special line bundles with degree ≥ 2g−6.
Abstract: We give necessary and sufficient conditions for non-special line bundles of degree2g — 2 and 2g — 3 being not normally generated. We also provide criteria for special line bundles of degreed > 2g — 6 being normally generated.
13 citations
TL;DR: In this article, the authors studied genus 5 curves with three bi-elliptic involutions by relating them to certain genus 3 curves and showed that there is only one such curve having exactly 24 Weierstrass points.
12 citations
11 citations
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54 citations
TL;DR: In this paper, the authors studied the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the Hilbert-Kunz multiplicity of non-singular curves in positive characteristic.
53 citations
36 citations
TL;DR: The notion of free linear systems on integral Gorenstein curves is introduced and it is obtained that each linear system on such a curve is obtained by enlarging the base scheme of a uniquely determined free linear system.
33 citations
TL;DR: The Weierstrass semigroup at all Fq2-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group.
Abstract: In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all \(\mathbb F_{q^2}\)-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.
32 citations