Akira Ohbuchi

Bio: Akira Ohbuchi is an academic researcher from University of Tokushima. The author has contributed to research in topics: Genus (mathematics) & Plane curve. The author has an hindex of 9, co-authored 36 publications receiving 170 citations.

On double coverings of a pointed non-singular curve with any Weierstrass semigroup

Abstract: Let H be a Weierstrass semigroup, ie, the set H(P) of integers which are pole orders at P of regular functions on C\ {P} for some pointed non-singular curve (C, P) In this paper for any Weierstrass semi group H we construct a double covering n : C --+ C with a ramification point P such that H(n(P)) = H We also de­ termine the semigroup H(P) Moreover, in the case where H starts with 3 we investigate the relation between the semigroup H(P) and the Weierstrass semigroup of a total ramification point on a cyclic covering of the projective line with degree 6

Automorphism group of plane curve computed by Galois points

Abstract: Let $$C \subset {\mathbb P}^2$$ be a smooth plane curve, and $$P_1, \ldots , P_m$$ be all inner and outer Galois points for $$C$$ . Each Galois point $$P_i$$ determines a Galois group at $$P_i$$ , say $$G_{P_i}$$ . Then, by the definition of Galois point, an element of the Galois group $$G_{P_i}$$ induces a birational transformation of $$C$$ . In fact, we see that it becomes an automorphism of $$C$$ . We call this an automorphism belonging to the Galois point $$P_i$$ . Then, we consider the group $$G(C)$$ generated by automorphisms belonging to all Galois points for $$C$$ . In particular, we investigate the difference between $$\mathrm{Aut} (C)$$ and $$G(C)$$ , so that we determine the structure of $$\mathrm{Aut} (C)$$ .

Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve ii

Abstract: Let H be a 4-semigroup, i.e., a numerical semigroup whose minimum positive element is four. We denote by 4r(H) + 2 the minimum element of H which is congruent to 2 modulo 4. If the genus g of H is larger than 3r(H) − 1, then there is a cyclic covering � : C −→ P 1 of curves with degree 4 and its ramification point P such that the Weierstrass semigroup H(P) of P is H (Komeda (1)). In this paper it is showed that we can construct a double covering of a hyperelliptic curve and its ramification point P such that H(P) is equal to H even if g ≤ 3r(H) − 1.

Normal generation of line bundles of high degrees on smooth algebraic curves

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.

Regularity on abelian varieties I

Abstract: We introduce the notion of Mukai regularity ($M$-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

Regularity on abelian varieties I

Abstract: We introduce the notion of Mukai regularity (M-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strenghtens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others make the subject of upcoming papers.

Equations of (1, d)-polarized abelian surfaces

Abstract: In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d ). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A via L ⊗n are all quadratic; a possible choice for a set of generators of IA are the Riemann theta relations. On the other hand, much less is known about embeddings via line bundles L of type (1, d ), that is line bundles L which are not powers of another line bundle on A. It is well-known that if d ≥ 5, and A is a general abelian surface, then L is very ample, while L can never be very ample for d < 5. However, even if d ≥ 5, L may not be very ample for special abelian surfaces. We will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface. A few special cases are well-documented in the literature: d = 4, in which case the general surface is a singular octic in P3, cf. [BLvS], and d = 5 in which case the abelian surface is described as the zero set of a section of the Horrocks-Mumford bundle [HM], whereas its homogeneous ideal is generated by 3 (Heisenberg invariant) quintics and 15 sextics (cf. [Ma]). Also, recent work by Manolache and Schreyer [MS] and by Ranestad [Ra] provides a description of the equations and syzygies in the case d = 7.

Regularity on abelian varieties II: Basic results on linear series and defining equations

Abstract: We apply the theory of M-regularity developed by the authors [Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), 285-302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called M-regularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-664]. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.