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Showing papers in "Tohoku Mathematical Journal in 1992"


Journal ArticleDOI

56 citations








Journal ArticleDOI
TL;DR: In this article, Tomari et al. showed that a 3D normal isolated singularity is a simple elliptic singularity if and only if the exceptional divisor of a β-factorial terminal modification is an irreducible normal Λ3-surface.
Abstract: A simple ^-singularity is a three-dimensional normal isolated singularity with a certain condition on the mixed Hodge structure on a good resolution. We prove here that a three-dimensional normal isolated singularity is a simple ^-singularity if and only if the exceptional divisor of a β-factorial terminal modification is an irreducible normal Λ3-surface. A simple ^-singularity is defined in terms of the Hodge structure as a threedimensional analogue of a simple elliptic singularity. It is well known that a simple elliptic singularity is characterized by the geometric structure of the minimal resolution (cf. [S], [II] and [Wl]). The aim of this paper is to prove that a simple ^-singularity is also characterized by the geometric structure of a (J-factorial terminal modification which is a three-dimensional analogue of the minimal resolution (cf. [M]). This characterization should help investigations of a simple A3-singularity which are being carried out from various viewpoints (cf. [T], [W2], [W3] and [Y]). The authors would like to thank Professors M. Tomari and K-i. Watanabe for their helpful advices during the preparation of this paper. Let/: X-+X be a good resolution of a normal isolated singularity (X, x), where a resolution is called a good resolution if E=f~ 1(x)τed is a divisor with normal crossings. We decompose £into irreducible components Ei{i=\,2,..., s). If (A", x) is a Gorenstein singularity, then we have a presentation of canonical divisors

18 citations


Journal ArticleDOI
TL;DR: In this paper, Morin constructed a new version of the halfway model for the eversion of the sphere, called the closed halfway model, whose image can readily be shown to be the set of zeros of an explicit polynomial of degree eight.
Abstract: Abstract. In this paper, I construct a new version of the halfway model for the eversion of the sphere, called the closed halfway model, whose image can readily be shown to be the set of zeros of an explicit polynomial of degree eight. For this purpose , a 4-parameter family of halfway models is thoroughly investigated. This family also contains the so-called open halfway model constructed in [A2]. The closed halfway model is chosen among the immersions of this family whose multiple loci contain two circles. Applied to the results of [A1], a similar study leads to notice that there exist Boy surfaces depending on two parameters, each of which intersects a given sphere along four circles (one parallel and three meridians). In the Appendix, Morin gives a coding in differential topological terms, of a sphere eversion which turns out to be minimal in many respects, so that, from now on, we no longer need to refer to pictures in order to present the subject.

17 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of determining when a generalized hypergeometric system becomes isomorphic under the condition that a certain associated aίfine toric variety is normal was studied.
Abstract: We treat the problem of shifting parameters of the generalized hypergeometric systems defined by Gelfand when their associated toric varieties are normal. In this context we define and determine the Bernstein-Sato polynomials for the natural morphisms of shifting parameters. We also give some examples. Let A = { χ ί 9 . . . , χN}^Z n be a finite subset with certain properties. In [G], [GGZ], [GZK1], [GZK2], [GKZ] and so on, Gelfand and his collaborators defined and studied generalized hypergeometric systems MΛ associated to A with parameter α. Aomoto defined and studied a broader class of systems (cf. [A1]-[A4]). Generalized hypergeometric systems of this kind were also defined in [KKM] and [H], where they were named canonical systems. For \\MΛ9 which corresponds to the differentiation of solutions. In this paper, we treat the problem of determining when/^. becomes isomorphic under the condition that a certain associated aίfine toric variety is normal. In § 1 and § 2, we define the system MΛ and the natural morphism fXj9 and give a necessary condition (Theorem 2.3) for the morphism fxj to be an isomorphism. In §3, we introduce an assumption, which we call the normality and keep throughout this paper. In §4, §5, and §6, we define an ideal B(χj) of the ^-functions for the morphism fχj9 and obtain a sufficient condition in terms of the 6-functions (Corollary 5.4) for the morphism fXj to be isomorphic. The ideal B(χj) turns out to be singly generated by a certain polynomial (Theorem 6.4). In §7, some example are given. The author would like to thank Professors Ryoshi Hotta and Masa-Nori Ishida for helpful conversation. 1. Generalized hypergeometric systems. First of all, we recall the definition of generalized hypergeometric systems following Gelfand et al. (cf. [GGZ]). Suppose we are given N integral vectors χ/ = (χι;,..., χnj)eZ n ( y = l , . . . , 7 V ) satisfying two conditions. (1) The vectors #!,. . . ,## generate the lattice Z. * Partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 33C70; Secondary 14M25, 16W50, 39B32.

11 citations









Journal ArticleDOI
TL;DR: For functional differential equations on a fading memory space, some relationships between the BC-stability and p-stabilities are studied in this paper, where it is shown that the BC total stability and BC-uniform asymptotic stability are respectively equivalent to the p-total stability and puniform stability.
Abstract: For functional differential equations on a fading memory space, some relationships between the BC-stabilities and p-stabilities are studied. Although the BC-uniform stability is weaker than the p-uniform stability, it is shown that the BC-total stability and BC-uniform asymptotic stability are respectively equivalent to the p-total stability and p-uniform asymptotic stability.




Journal ArticleDOI
Kazuhiro Konno1
TL;DR: In this paper, the moduli space is non-reduced in many cases and the rational map associated with a semi-canonical bundle induces a linear pencill of nonhyperelliptic curves of genus three.
Abstract: We classify even canonical surfaces on the Castelnuovo lines, and show that the moduli space is non-reduced in many cases. We show that, in most cases, the rational map associated with a semi-canonical bundle induces a linear pencill of nonhyperelliptic curves of genus three, and that a nonsingular rational curve with self-intersection number —2 appears as a fixed component of the semi-canonical system. By the latter, we can apply a result of Burn and Wahl to show that they are obstructed surfaces. Introduction. According to [8], we call a minimal surface a canonical surface if the canonical map induces a birational map onto its image. Canonical surfaces with c\ = 3pg — l and 3pg — β were studied in our previous papers [1] and [10] (see also [4] and [8]). These are regular surfaces whose canonical linear system | K\ has neither fixed components nor base points. In this article, we list up those which are even surfaces in order to supplement [1] and [10]. Here, we call a compact complex manifold of dimension 2 an even surface if its second Steifel-Whitney class w2 vanishes ([8]). This topological condition implies the existence of a line bundle L with K=2L. In a recent paper [9], Horikawa classified all the even surfaces with/^ = 10, # = 0 and K = 24 (numerical sextic surfaces). Following [9], we consider the rational map ΦL associated with \L\ also in the remaining cases. Recall that most canonical surfaces with cj = 3pg — Ί, 3pg — 6 have a pencil | Z> | of nonhyperelliptic curves of genus 3. Therefore, it is naturally expected that ΦL should be composed of such a pencil. We show that this is the case, except for numerical sextic surfaces. Let/ : S-+P be the corresponding fibration. It turns out that the fact that S is an even surface forces f*Θ{K) to be very special (Lemmas 1.2 and 2.2). Using this, we can determine the fixed part Z of | L |. The remaining problem is to write down the equation of the canonical model. When K = 3pg — 1, we have no difficulty in doing this, since the (relative) canonical image itself is the canonical model. On the other hand, when K = 3pg — 6, we need to study the bi-graded ring 0// o (αZ) + jSZ) as in [9]. The calculation after Lemma 2.3 is a verbatim translation of [9]. As a by-product, we find that the moduli space is non-reduced in many cases (Theorems 1.5 and 2.5). The point is the presence of a ( —2)-curve contained in Z. Then a general result of Burns and Wahl [3] can be applied to show that the Kuranishi space is everywhere singular. As far as surfaces of general type are concerned, such pathological 1991 Mathematics Subject Classification. Primary 14J29; Secondary 14J15.




Journal ArticleDOI
TL;DR: In this article, the authors classify hypersurfaces in a Euclidean space which admit isometric deformations preserving mean curvature, under some conditions, and show that the deformations preserve mean curvatures.
Abstract: Under some conditions we classify hypersurfaces in a Euclidean space which admit isometric deformations preserving mean curvature.



Journal ArticleDOI
TL;DR: In this article, the authors studied the distribution of rational points on a hyperelliptic surface defined over an algebraic number field, and showed that this distribution is very similar to the rational points distribution on an abelian surface, and that a conjecture of Batyrev-Manin holds for such a surface.
Abstract: In this paper, we study distribution of rational points on a hyperelliptic surface defined over an algebraic number field, and show that this distribution is very similar to the distribution of rational points on an abelian surface. As an application, we show that a conjecture of Batyrev-Manin holds for such a surface.

Journal ArticleDOI
TL;DR: In this paper, the authors obtained generalizations of the celebrated theorem of Malmquist-Yosida for second order differential equations with admissible meromorphic solutions with the aid of Nevanlinna theory.
Abstract: We treat second order differential equations which have admissible meromorphic solutions. With the aid of Nevanlinna theory, we obtain generalizations of the celebrated theorem of Malmquist-Yosida.