Updated on 2025/07/02

写真a

 
KAJI, Hajime
 
Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor
Degree
Master of Science ( Waseda University )
Doctor of Science ( Waseda University )
Homepage URL
http://pc193097.pc.waseda.ac.jp/

Research Experience

  • 1999
    -
    　

    Waseda University, Professor

  • 1993
    -
    1999

    Waseda University, Associate Professor

  • 1989
    -
    1993

    Yokohama City University, Research Assistant

  • 1987
    -
    1989

    Waseda University, Research Assistant

Committee Memberships

  • 2015.09
    -
    　

    日本数学会  代数学分科会運営委員

Professional Memberships

  • 　
    　
    　

    日本数学会

Research Areas

  • Geometry / Algebra

Research Interests

  • Algebraic Geometry

 

Papers

  • Degree formula for Grassmann bundles

    Hajime Kaji, Tomohide Terasoma

    JOURNAL OF PURE AND APPLIED ALGEBRA   219 ( 12 ) 5426 - 5428  2015.12  [Refereed]

    　View Summary

    Let X be a non-singular quasi-projective variety over a field, and let epsilon be a vector bundle over X. Let G(X)(d, epsilon) be the Grassmann bundle of epsilon over X parametrizing corank d subbundles of epsilon with projection pi : G(X)(d,epsilon) -> X, let Q <- pi*epsilon be the universal quotient bundle of rank d, and denote by theta the Plucker class of G(X)(d, epsilon), that is, the first Chern class of the Plucker line bundle, det Q. In this short note, a closed formula for the push-forward of powers of the Plucker class theta is given in terms of the Schur polynomials in Segre classes of epsilon, which yields a degree formula for G(X)(d, epsilon) with respect to theta when X is projective and Lambda(d)epsilon is very ample. (C) 2015 Elsevier B.V. All rights reserved.

    DOI

    Scopus

    4
    Citation
    (Scopus)

  • Projective varieties admitting an embedding with Gauss map of rank zero

    Satoru Fukasawa, Katsuhisa Furukawa, Hajime Kaji

    ADVANCES IN MATHEMATICS   224 ( 6 ) 2645 - 2661  2010.08  [Refereed]

    　View Summary

    We introduce an intrinsic property for a projective variety as follows there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties. In fact. using (GMRZ). we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characterisitic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ). and show that a general hypersurfaces of low degree does not satisfy (GMRZ). (C) 2010 Elsevier Inc All rights reserved.

    DOI

    Scopus

    1
    Citation
    (Scopus)

  • The separability of the Gauss map and the reflexivity for a projective surface

    Satoru Fukasawa, Hajime Kaji

    MATHEMATISCHE ZEITSCHRIFT   256 ( 4 ) 699 - 703  2007.08  [Refereed]

    　View Summary

    It is known that if a projective variety X in P-N supercript stop is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

    DOI

    Scopus

    8
    Citation
    (Scopus)

  • The classification of orbits by a natural action of certain reductive linear groups

    H. Kaji, O. Yasukura

    Yokohama Math. J.   53   39 - 61  2006  [Refereed]

  • Projective geometry of Freudenthal's varieties of certain type

    H Kaji, O Yasukura

    MICHIGAN MATHEMATICAL JOURNAL   52 ( 3 ) 515 - 542  2004  [Refereed]

  • Adjoint varieties and their secant varieties

    H Kaji, M Ohno, O Yasukura

    INDAGATIONES MATHEMATICAE-NEW SERIES   10 ( 1 ) 45 - 57  1999.03  [Refereed]

    　View Summary

    The purpose of this article is to show how the graded decomposition of complex simple Lie algebras g can be applied to studying adjoint varieties X and their secant varieties Sec X. Firstly quadratic equations defining adjoint varieties are explicitly given. Secondly it is shown that dim Sec X = 2 dim X for adjoint varieties X in two ways: one is based on Terracini's lemma, and the other is on some explicit description of Sec X in terms of an orbit of the adjoint action. Finally it is shown that the contact loci of the secant variety to its embedded tangent space have dimension two if X is adjoint.

  • Homogeneous projective varieties with degenerate secants

    H Kaji

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   351 ( 2 ) 533 - 545  1999.02  [Refereed]

    　View Summary

    The secant variety of a projective variety X in P, denoted by Sec X, is defined to be the closure of the union of lines in P passing through at least two points of X, and the secant deficiency of X is defined by delta := 2 dim X + 1 - dim Sec X. Mie list the homogeneous projective varieties X with delta > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety X with SecX not equal P and delta > 8, and the Es-variety is the only homogeneous projective variety with largest secant deficiency delta = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.

  • On the inseparable degrees of the Gauss map and the projection of the conormal variety to the dual of higher order for space curves

    H KAJI

    MATHEMATISCHE ANNALEN   292 ( 3 ) 529 - 532  1992.03  [Refereed]

  • ON THE INSEPARABLE DEGREE OF THE GAUSS MAP OF HIGHER-ORDER FOR SPACE CURVES

    M HOMMA, H KAJI

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   68 ( 1 ) 11 - 14  1992.01  [Refereed]

    　View Summary

    Let X be a curve non-degenerate in a projective space P(N) defined over an algebraically closed field of positive characteristic p, consider the Gauss map of order m defined by the osculating m-planes at general points of X, and denote by {b}0 less-than-or-equal-to j less-than-or-equal-to N the orders of X. We prove that the inseparable degree of the Gauss map of order m is equal to the highest power of p dividing b(m + 1).

  • Strangeness of higher order for space curves

    H KAJI

    COMMUNICATIONS IN ALGEBRA   20 ( 6 ) 1535 - 1548  1992  [Refereed]

  • ON THE GAUSS MAPS OF SPACE-CURVES IN CHARACTERISTIC-P .2.

    H KAJI

    COMPOSITIO MATHEMATICA   78 ( 3 ) 261 - 269  1991.06  [Refereed]

  • ON THE GAUSS MAPS OF SPACE-CURVES IN CHARACTERISTIC-P

    H KAJI

    COMPOSITIO MATHEMATICA   70 ( 2 ) 177 - 197  1989.05  [Refereed]

  • Example of σ-transition matrices defining the horrocks-mumford bundle

    Hajime Kaji

    Tokyo Journal of Mathematics   12 ( 1 ) 21 - 32  1989  [Refereed]

    DOI

    Scopus

    3
    Citation
    (Scopus)

  • ON THE TANGENTIALLY DEGENERATE CURVES

    H KAJI

    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES   33   430 - 440  1986.06  [Refereed]

  • On the normal bundles of rational space curves

    H KAJI

    MATHEMATISCHE ANNALEN   273 ( 1 ) 163 - 176  1985  [Refereed]

  • On the tangentially degenerate curves, II

    Hajime Kaji

    BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY   45 ( 4 ) 745 - 752  2014.12  [Refereed]

    　View Summary

    A projective curve X in a"(TM) (N) is said to be tangentially degenerate if a tangent line to X at a general point P meets X again at a point other than P. In this article, we prove that if X is tangentially degenerate in characteristic zero, then X is contained in a 2-plane under a certain condition on singularities of X. We also investigate a counter-example of E. Esteves and M. Homma in positive characteristic case, and present a conjecture that our investigation leads to.

    DOI

    Scopus

    9
    Citation
    (Scopus)

  • Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

    Satoru Fukasawa, Hajime Kaji

    JOURNAL OF PURE AND APPLIED ALGEBRA   214 ( 3 ) 297 - 300  2010.03  [Refereed]

    　View Summary

    We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p > 0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p = 2. (C) 2009 Elsevier B.V. All rights reserved.

    DOI

    Scopus

    4
    Citation
    (Scopus)

  • The separability of the Gauss map versus the reflexivity

    Hajime Kaji

    GEOMETRIAE DEDICATA   139 ( 1 ) 75 - 82  2009.04  [Refereed]

    　View Summary

    In projective algebraic geometry, various pathological phenomena in positive characteristic have been observed by several authors. Many of those phenomena concerning the behavior of embedded tangent spaces seem to be controlled by the separability of (the extension of function fields defined by) the Gauss map, or by the reflexivity with respect to the projective dual for a projective variety. The purpose of this paper is to survey the studies on the relationship between the separability of the Gauss map and the reflexivity for a projective variety: Is the separability of the Gauss map equivalent to the reflexivity for a projective variety?.

    DOI

    Scopus

    3
    Citation
    (Scopus)

  • The reflexivity of a Segre product of projective varieties

    Satoru Fukasawa, Hajime Kaji

    MATHEMATISCHE ANNALEN   342 ( 2 ) 279 - 289  2008.10

    　View Summary

    We study the reflexivity of a Segre product of a projective space P-m and a projective variety Y, and give a criterion for P-m x Y to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.

    DOI

    Scopus

  • Existence of a non-reflexive embedding with birational Gauss map for a projective variety

    Satoru Fukasawa, Hajime Kaji

    MATHEMATISCHE NACHRICHTEN   281 ( 10 ) 1412 - 1417  2008  [Refereed]

    　View Summary

    We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in P-N to re-embed into some projective space P-M so as to be non-reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in P-N is identically zero; hence the projective variety X re-embedded in P-M yields a negative answer to Kleiman-Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in PN with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in P-N satisfying the assumption. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Woinheim.

    DOI

    Scopus

    5
    Citation
    (Scopus)

  • 公開鍵暗号を解読せよ!—君もスパイになれる?—

    楫 元

    日本数学会「数学通信」   10 ( 2 ) 5 - 37  2005.08

    CiNii

  • エンピツ片手にテキストを読もう[ブックガイド]代数

    楫 元

    数学セミナー/日本評論社   42 ( 8 ) 34 - 35  2003.08

  • On the duals of Segre varieties

    H Kaji

    GEOMETRIAE DEDICATA   99 ( 1 ) 221 - 229  2003.06  [Refereed]

    　View Summary

    The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?

  • Outside In (Delle Maxwell, Silvio Levy, Tamara Munzner監督; Minnesota Geometry Center制作)

    楫 元

    数学セミナー/日本評論社   39 ( 9 ) 85 - 85  2000.09

  • Tangent loci and certain linear sections of adjoint varieties

    H Kaji, O Yasukura

    NAGOYA MATHEMATICAL JOURNAL   158   63 - 72  2000.06  [Refereed]

    　View Summary

    An adjoint variety X(G) associated to a complex simple Lie algebra g is by definition a projective variety in P*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of sit-triples. Secondly for a graded decomposition of contact type g = +(-2 less than or equal to i less than or equal to 2) g(i), we show that the intersection of X(8) and the linear subspace P*(g(1))in P*(g) coincides with the cubic Veronese variety associated to g.

  • Secant varieties of adjoint varieties: Orbit decomposition

    H Kaji, O Yasukura

    JOURNAL OF ALGEBRA   227 ( 1 ) 26 - 44  2000.05  [Refereed]

    　View Summary

    The orbit decomposition of secant varieties of adjoint varieties is given. (C) 2000 Academic Press.

  • アレフゼロ・ホテル?無限ホテル業界の軌跡?

    楫 元

    数学セミナー/日本評論社   38 ( 9 ) 27 - 31  1999.08

  • Secant varieties of adjoint varieties

    H.Kaji

    Matem?tica Contempor?nea   14   75 - 87  1998  [Refereed]

  • On the generic injectivity of the Gauss map in positive characteristic

    H Kaji, A Noma

    JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK   482   215 - 224  1997  [Refereed]

    　View Summary

    The purpose of this article is to give a criterion for Gauss maps of projective varieties in positive characteristic to be generically injective. As its applications, we show that the following have generically injective Gauss maps for any embeddings: smooth projective varieties with ample cotangent bundles, smooth subvarieties of products of smooth curves with genus at least 2, smooth subvarieties of abelian varieties with ample normal bundles, and smooth projective varieties of non-zero top Chern classes with separable finite morphisms to abelian varieties.

  • 標数pの世界

    楫 元

    数理科学/サイエンス社   369  1994.03

  • On the space curves with the same image under the Gauss map

    H KAJI

    MANUSCRIPTA MATHEMATICA   80 ( 3 ) 249 - 258  1993.09  [Refereed]

    　View Summary

    From an irreducible complete immersed curve X in a projective space P other than a line, one obtains a curve X' in a Grassmann manifold G of lines in P that is the image of X under the Gauss map, which is defined by the embedded tangents of X. The main result of this article clarifies in case of positive characteristic what curves X have the same X': It is shown that X is uniquely determined by X' if X, or equivalently X', has geometric genus at least two, and that for curves X1 and X2 with X1 not-equal X2 in P, if X1' = X2' in G and either X1 or X2 is reflexive, then both X1 and X2 are rational or supersingular elliptic; moreover, examples of smooth X1 and X2 in that case are given.

  • On the space curves with the same dual variety

    H KAJI

    JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK   437   1 - 11  1993  [Refereed]

  • Characterization of space curves with inseparable Gauss maps in extremal cases

    H KAJI

    ARCHIV DER MATHEMATIK   58 ( 6 ) 539 - 546  1992.06  [Refereed]

  • ON THE VECTOR BUNDLES WHOSE ENDOMORPHISMS YIELD AZUMAYA ALGEBRAS OF CYCLIC TYPE

    H KAJI

    JOURNAL OF ALGEBRA   117 ( 2 ) 297 - 326  1988.09  [Refereed]

  • On tangentially non-degenerate plane curves

    S. Arima, H. Kaji

    Bull. Sci. Engrg. Res. Lab. Waseda Univ.   111   43 - 43  1985  [Refereed]

▼display all

Books and Other Publications

  • 数学ガイダンスhyper

    数学セミナー編集部

    日本評論社  2005.03 ISBN: 9784535784277

  • 工科系のための初等整数論入門?公開鍵暗号をめざして?

    楫 元

    培風館  2000.07 ISBN: 9784563014858

Presentations

  • Pieri’s Formula and Its Applications

     [Invited]

    Presentation date： 2024.03

  • Some remarks on degree formulae for 2-step flag varieties

     [Invited]

    Presentation date： 2023.12

  • Push-Forward Formula for Grassmann Bundles and Its Application

    Presentation date： 2023.10

  • Some remarks on degree formulae for 2-step flag varieties ―やっぱり Borel-Hirzebruch は偉い!!―

     [Invited]

    Presentation date： 2023.09

  • Higher Gauss maps of Veronese varieties ―a generalization of Boole’s formula―

     [Invited]

    Presentation date： 2019.12

  • Two Results on Curves in P^3

    Presentation date： 2019.11

  • Degree Formula for Grassmann Bundles

     [Invited]

    NSTS Seminar in Algebraic Geometry 

    Presentation date： 2019.10

  • Powers of Ideals

    第二回宇都宮大学代数幾何研究集会 

    Presentation date： 2019.09

  • Degree formula for Grassmann bundles, II

     [Invited]

    研究集会「Arithmetic and Algebraic Geometry 2019 ---in honour of Professor Tomohide Terasoma's 60th birthday---」 

    Presentation date： 2019.01

  • Degree Formulae for Two-Step Flag Varieties

    研究集会「第一回宇都宮大学代数幾何研究集会」 

    Presentation date： 2018.08

  • Degree formula for Grassmann bundles and its applications

     [Invited]

    研究集会「ベクトル束の分裂・構成・安定性とその応用」 

    Presentation date： 2018.06

  • グラスマン束の次数公式 (新証明)

     [Invited]

    第25回沼津改め静岡研究会 

    Presentation date： 2018.03

  • On a problem posed by Alessandro Terracini

     [Invited]

    第５回代数幾何学研究集会－宇部－ 

    Presentation date： 2018.01

  • グラスマン束の次数公式 (新証明) とその応用

     [Invited]

    射影多様体の幾何とその周辺 

    Presentation date： 2016.10

    Event date：
    2016.10
    　
    　

  • グラスマン束の次数公式 (新証明) とその応用

    楫元

    山形代数幾何小研究集会 

    Presentation date： 2016.08

    Event date：
    2016.08
    　
    　

  • グラスマン束の次数公式と高次ガウス写像への応用

     [Invited]

    Presentation date： 2015.03

  • Higher Gauss maps of Veronese varieties---a generalization of Boole's formula---

    KAJI, Hajime  [Invited]

    Mini-conference on Algebraic Geometry  (National Taiwan University) 

    Presentation date： 2015.03

  • グラスマン束の次数公式(続)

     [Invited]

    Presentation date： 2014.09

  • グラスマン束の次数公式

    Presentation date： 2014.03

  • Degree formula for Grassmann bundles

     [Invited]

    Symposium on Projective, Algebraic Varieties and Moduli 2014 (in honor of Professor Changho Keem's 60th birthday), Seoul National University (Seoul, Korea). 

    Presentation date： 2014.02

  • グラスマン束の次数公式

    Presentation date： 2013.12

  • 接的退化曲線について

    Presentation date： 2013.04

  • A tangential trisecant lemma

    研究集会, ALGA-2012, the 12th Meeting of the Brazilian Group in Commutative Algebra and Algebraic Geometry, IMPA (Rio de Janeiro, Brazil). 

    Presentation date： 2012.08

  • A tangential trisecant lemma

    Presentation date： 2012.03

  • A tangential trisecant lemma

    Symposium on Projective Algebraic Varieties and Moduli, Novotel Ambassador (Busan, Korea) 

    Presentation date： 2012.02

  • Homogeneous projective varieties with unique secant property

    Presentation date： 2011.11

  • 随伴多様体の射影幾何的魅力

    Presentation date： 2011.03

  • Gauss maps in positive characteristic

    Algebra Seminar, Seoul National University (Seoul, Korea). 

    Presentation date： 2010.09

  • Gauss maps in positive characteristic

    研究集会, ALGA-2010, the Tenth Meeting of the Brazilian Group in Commutative Algebra and Algebraic Geometry, IMPA (Rio de Janeiro, Brazil). 

    Presentation date： 2010.07

  • Gauss maps in positive characteristic

    Workshop ''2010 Algebra and Geometry of Subvarieties of Projective Space (in honor of Professor Fyodor Zak on his 60th birthday),'' KAIST (Daejeon, KOREA). 

    Presentation date： 2010.01

  • Projective varieties admitting an embedding with Gauss map of rank zero

    Presentation date： 2009.11

  • Gauss maps in positive characteristic

    Presentation date： 2009.09

  • Orbit decomposition of secant varieties of adjoint varieites

    Presentation date： 2009.09

  • Any algebraic variety in positive characteristic admits a projective model with inseparable Gauss map (深澤知氏との共同講演)

    日本数学会秋期総合分科会, 大阪大学. 

    Presentation date： 2009.09

  • Projective varieties admitting an embedding with Gauss map of rank zero (深澤知氏, 古川勝久氏との共同講演),

    Presentation date： 2009.09

  • The generic smoothness of the Gauss map versus the reflexivity

    Presentation date： 2009.02

  • The generic smoothness of the Gauss map versus the reflexivity

    Presentation date： 2008.10

  • Existence of a non-reflexive embedding with birational Gauss map for a projective variety

    Presentation date： 2008.09

  • The reflexivity of a Segre product of projective varieties

    Presentation date： 2008.09

  • Projective varieties with degenerate secant varieties

    Presentation date： 2008.06

  • Existence of a nonreflexive embedding with birational Gauss map for a projective variety

    International Conference "IV Iberoamerican Conference on Complex Geometry", (Brasil, Ouro Preto) 

    Presentation date： 2007.08

  • Projective geometry of adjoint varieties

    Algebraic Geometry Seminar, Seoul National University (Seoul, Korea) 

    Presentation date： 2007.06

  • Projective varieties with degenerate secant varieties

    Colloquium, National Institute for Mathematical Sciences (Daejeon, Korea) 

    Presentation date： 2007.06

  • Projective varieties with degenerate secant varieties

    Algebraic Geometry Seminar, Seoul National University (Seoul, Korea) 

    Presentation date： 2007.06

  • The separability of the Gauss map and the reflexivity for a projective surface

    Presentation date： 2006.09

  • 随伴多様体の射影幾何的魅力

    研究集会「Symplectic Geometry とその周辺」, 岐阜経済大学 

    Presentation date： 2005.11

  • 随伴多様体の射影幾何的魅力

    研究集会「部分多様体の微分幾何学」, 京都大学数理解析研究所 

    Presentation date： 2005.06

  • ある種のフロイデンタール多様体の射影幾何

    シンポジウム「代数曲線論」, 神奈川工科大学 

    Presentation date： 2004.12

  • 公開鍵暗号を解読せよ!−君もスパイになれる?−

    湘南数学セミナー, 湘南国際村センター (神奈川県三浦郡葉山町) 

    Presentation date： 2004.12

  • 素数を見つけて百万長者になろう!?

    現代数学入門市民講座, 湘南国際村センター (神奈川県三浦郡葉山町) 

    Presentation date： 2004.12

  • フロイデンタール多様体の射影幾何

    城崎代数幾何シンポジウム 

    Presentation date： 2004.10

  • ガウス写像の非分離性に関するKleiman-Pieneの問題について

    高知研究集会「射影多様体の幾何とその周辺」 

    Presentation date： 2003.08

  • 正標数の代数多様体のガウス写像について

    談話会、東北大学 

    Presentation date： 2003.06

  • reflexivityとGauss射の分離性に関する Kleiman-Piene の問題について

    代数幾何学シンポジウム ---高次元多様体、正標数上の話題を中心として--- 

    Presentation date： 2003.01

  • Algebras with ternary product and projective algebraic geometry (三項演算をもつ代数系と射影代数幾何)

    城崎代数幾何シンポジウム 

    Presentation date： 2001.10

  • Secant varieties of adjoint varieties---orbit decomposition---

    ミニワークショップ 「代数多様体と複素解析特異点」 

    Presentation date： 1999.12

  • Secant varieties of adjoint varieties---orbit decomposition---

    研究集会, 「リー群と幾何学」 

    Presentation date： 1998.12

  • Degeneration of secant varieties

    研究集会, 「部分多様体と特異点のトポロジーと幾何」 

    Presentation date： 1998.11

  • Secant varieties of adjoint varieties

    研究集会「School on Commutative Algebra and Projective Varieties」 (長野メルパルク) 

    Presentation date： 1998.03

  • Secant varieties of K?hler C-spaces of Boothby type

    研究集会「等質空間と Hesse 構造の幾何とその周辺」 (山口大学) 

    Presentation date： 1997.12

  • On the pure inseparability of the Gauss maps

    日本数学会代数学分科会 

    Presentation date： 1993.03

  • On

    日本数学会代数学分科会 

    Presentation date： 1992.04

  • On the inseparable degree of the Gauss map of higher order for space curves

    日本数学会代数学分科会 

    Presentation date： 1991.10

  • Strangeness of higher order for space curves

    日本数学会代数学分科会 

    Presentation date： 1991.10

  • Characterization of space curves with inseparable Gauss maps in extremal cases

    日本数学会代数学分科会 

    Presentation date： 1990.09

  • On the Gauss maps of space curves in characteristic $p$

    日本数学会代数学分科会 

    Presentation date： 1989.09

  • Gauss maps of space curves in characteristic $p$

    都立大学研究集会 ``解析および代数多様体上の諸問題'' 

    Presentation date： 1988.01

  • On the vector bundles whose endomorphisms yield quaternion algebras over a product of elliptic curves

    日本数学会代数学分科会 

    Presentation date： 1987.10

  • On the space curves with separable Gauss map

    日本数学会代数学分科会 

    Presentation date： 1987.10

  • On the space curves with purely inseparable Gauss map

    日本数学会代数学分科会 

    Presentation date： 1987.10

  • Example of $sigma$-transition matrices defining the Horrocks-Mumford bundle

    日本数学会代数学分科会 

    Presentation date： 1987.10

  • On the vector bundleds whose endomorphisms yield quaternion algebras over a product of elliptic curves

    第9回可換環論シンポジウム 

    Presentation date： 1987.10

  • On the vector bundles whose endomorphisms yield Azumaya algebras of cyclic type

    京都大学数理解析研究所研究集会 ``Analytic varieties および stratified space における諸問題'' 

    Presentation date： 1986.10

  • On the normal bundles of rational space curves

    日本数学会代数学分科会 

    Presentation date： 1985.09

  • On the tangentially degenerate curves

    日本数学会代数学分科会 

    Presentation date： 1985.04

  • Jumping-conics of vector bundles

    日本数学会代数学分科会 

    Presentation date： 1983.04

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Research Projects

  • Projective Algebraic Geometry in Positive Characteristic

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2016.04
    -
    2020.03
     

    KAJI Hajime

    　View Summary

    In 1845, George Boole discovered a formula for the degree of dual manifolds of Veronese varieties. In general, the dual variety of a projective variety X is a closed variety of dual projective space given by the closure of the set of hyperplanes tangent to X. In the previous research, for Veronese varieties, we obtained an degree formula (unpublished) that generalizes dual varieties into images of general Gauss maps. In this research, we investigated the asymptotic behavior of the image degree of the general Gauss map of the Veronese varieties using the generalized degree formula.

  • Projective Algebraic Geometry in Positive Characteristic

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2013.04
    -
    2016.03
     

    KAJI Hajime, FUKASAWA Satoru, FURUKAWA Katsuhisa

    　View Summary

    I obtained a degree formula for Grassmann bundles associated to vector bundles on non-singular projective varieties, as a joint work with Tomohide TERASOMA. I wrote a joint paper with TERASOMA for the resuls, and published it in a scientific journal. I also studied the higher Gauss maps for projective varieties, and obtained a degree formula for the images of higher Gauss maps of Veronese varieties, which yields a generalization of the classical formula by Boole. I wrote a paper for the result of a continued work on the linear degeneration of tangentially degenerate curves, and published it in a scientific journal

  • Projective Algebraic Geometry in Positive Characteristic

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2010.04
    -
    2013.03
     

    KAJI Hajime, FUKASAWA Satoru, FURUKAWA Katsuhisa

    　View Summary

    I obtained a few results on non-singular projective varieties with Gauss map of rank zero, on the linearity of Gauss fibres, and on the linear degeneration of tangentially degenerate curves. I also organized a symposium “Miyako no Seihoku Algebraic Geometry Symposium.

  • On secant varieties of projective varieties

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2007
    -
    2008
     

    KAJI Hajime

  • Real analytic approach to the stability theory of nonlinear evolution equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2000
    -
    2003
     

    SHIBATA Yoshihiro, YAMAZAKI Masao, KAJI Hajime, TANAKA Kazunaga, KOBAYASHI Takayuki, SHIMIZU Senjo

    　View Summary

    1.Stability of the Oseen flow in the n-dimensional exterior domain (n>2).
    2.Stability of the Couette flow and the Poiseuille flow in the infinite layer.
    3.Rate of convergence of the non-stationary flow to the stationary flow of compressible viscous fluid.
    4.Resolvent estimate of solutions to the Stokes equation with Neumann boundary condition.

  • Research on homogeneous projective varieties by Lie algebra and algebraic geometry

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1998
    -
    2001
     

    YASUKURA Osami, MAEDA Hidetoshi

    　View Summary

    H. Maeda gave the following results :(1) Let E be an ample vector bundle of rank n-2 on a complex projective manifold of dimension n having a section whose zero locus Z is an algebraic surface of Kodaira dimension 1. Then the structure of E is completely determined. This generalizes Sommese and Shepherd-Barron's results on ample divisors.(2) A classification of the polarized varieties (X, E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle E of rank n-1 on X such that E has a section whose zero locus is a smooth elliptic curve. And the property of E is investigated when E is very ample having a section whose zero locus equals a hyperelliptic curve of genus non less than two.(3) In particular, a classification of such (X, E)'s is given when the genus of Z equals two. A classification of the polarized varieties (X, E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle E of rank n-r on X such that E has a section whose zero locus Z is a smooth r-dimensional submanifold of X when Z contains a bielliptic curve section.O. Yasukura, in collaboration with H. Kaji (Waseda Univ., Japan and IMPA, Brasil), gave a concrete investigation on the relations among three objects : the adjoint varieties, symplectic triple systems and the gradation of contact type for complex simple Lie algebras. And they described and proved projective geometric properties on Freudenthal varieties in terms of the concept,of symplectic triple systems. In particular, for the adjoint varieties, the orbit decomposition and projective geometric description of the secant varieties are given. For Freudenthal varieties, the linear sectional relation with the corresponding adjoint varieties and an essential proof for the homogeneity are obtained as well as several other proves

  • Boundary conditions for gavge coupled Dirac operators and their invariants.

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1997
    -
    1998
     

    KORI Toshiaki, YONIDA Gen, SUZUKI Takeru, KOSAMA Tokitake, TANAKA Kazunaga, KAJI Hajime

    　View Summary

    (1) T.Kori investigated the theory of the index of a gauge coupled Dirac operator with Grassmannian boundary condition, especially he gave a direct method of calculations not using the Atiyah-Patodi-Singer theory for those problems on the four dimensional hemisphere.(2) T.Kori.proved a formula about the chiral anomaly of gauge coupled Dirac operators. Here he proved that the index of a gauge coupled Diracoperator on S^4 is equal to the index of the geometric Dirac operator on the hemisphere with a Grassmannian boundary condition comming from the vector potential, that is, the effect by the gauge is absorbed in the boundary condition. This result will be published in the Proceeding of the conference on Geometric Aspects of Partial Differential Equations as one volume of AMS Contemporary Mathematics series.(3) The problem of extension of spinors from the boundary to the interior or the exterior as zero mode spinors is solved. By this an analogy of the Laurent expansion theorem for zero mode spinors is obtained. Thus the concepts of meromorphic spinors and their residues are introduced. He proved the residue theorem on a domain in S^4. Many theorems that are counterparts of what are known in complex function theory are expected to hold in our framework of spinor analysis. This will be our next project

  • 場の量子論の数学的研究

    　View Summary

    本研究の目的の一つは,ChernーSimons理論の運動方程式と古典的なsoliton方程式(戸田方程式,非線形Schrodinger方程式等)との関係を明らかにすることであった。我々は一連の研究の中で(1)Jackiw達が研究したselfーdual ChernーSimons solitonsと(2)非線形Grassmann σーmodelsの古典解と(3)戸田方程式(及び連続な戸田方程式)の古典解の3者の間の関係をほぼ明らかにした。即ち,これらには(☆)Lie環sl(2;〓)のsl(n;〓)へのspin(n-1)/2の既約表現(highest weight表現)が共通して内在していることを明らかにした。これらは古典論である。これらの量子論への応用(又は量子論での対応物)が期待される。この場合、上述の(☆)は(☆☆)sl(n;〓)→Uirasora algebraに変わるものと思われる。次年度の研究課題としたい

  • 量子群の作用素環論的研究

    　View Summary

    量子群SUi(n)のIII_1型PowersファクターRiへの無限テンソル積型作用に関する不動点環は、Janes射影列により生成されるAFDII_1型ファクターに成り、Ri上のPowers状態を、このII_1型ファクターへ制限したものはMarkovトレイスになる。しかも、このトレイスを用いると、n=2の場合には1変数の、n≧3の場合には2変数のJanes多項式が与えられる。これらの事柄は、比較的早い時期から、証明も無いまゝに、専門家の間では良く知られていた。証明が無かった理由の1つは、量子群の作用素環への作用を考える場合には、量子群を作用素環を用いて記述する必要が生じるが、その準備ができていなかったからである。そこで、量子群の座標環であるHopf*環を稠密*部分多元環として含み、しかも、Hopf*環の群構造をその上へ拡張できるような作用素環の枠組を作る必要がある。しかし、これを具体化しようと思うと、余積の値域の問題、全逆写像の非有異性の問題など、取り扱いが容易でないと思われる問題とすぐに直面する。他方、作用素環では量子群が発見される以前から、群の量子化としてKac環という対象がvon Neumaun環を用いて定義され、研究されてきた。そこで、この考え方や枠組を参考に、量子群の9-変形に対応した、Kac環の9変形に相当する、Woronowiez環なるものを定義し、局所コンパクト群の場合に知られている、Portrjayin-淡中-Krein-辰馬の双対定現に対応する命題を示した。このWoronowiez環は、Hopf環の場合と違って、冨田-竹崎理論を用いて一般的に定義されているため、具体的な量子群がこの定義に適合するかどうかの確認が必要で、現在のところ、この適分性が確認できた量子群はSUi(n)以外には無い。最初に述べた結果の証明にはこれで充分であり、これでようやく、量子群SUi(n)のvon Neumaun環への作用を考えることができるようになった

  • 代数多様体の射影空間への埋め込みに関する研究

    　View Summary

    射影多様体M⊆Pに対して、Mの一般の点xにMのxでのPに埋め込まれた接空間T_xMを対応させることにより得られるMからグラスマン多様体への有理写像を、射影多様体Mのガウス射という。平成6年度の研究目的は、正標数の代数閉体上定義された射影多様体のガウス射の構造を調べることであった。M⊆Pに対して、法束の双対から微分の2階対称積への自然な写像N^V_<M/P>→S^2Ω^1_Mが定まる。これはガウス射の微分を表わすもとして知られ、その像(の基底)は第2基本形式と呼ばれている。Mの各点xで第2基本形式は、xでのZariski接空間t_xMの1次元部分空間の全体のなす射影空間P_*(t_xM)上の2次のlinear systemを定めることが容易に分かるが、今年度はこのlinear system、特にそのbase locusと射影多様体の性質との関係について研究を行った。射影多様体M⊆Pの一般の点xに対してM∩T_xM={x}が成立するとき、Mを接的に非退化合ですら未解決であり、本研究課題おいても非常に重要と思われる。今年度の研究を通じて、その分類に対しても第2基本形式を調べることが有効であることが判ってきた。将来は、この問題についても研究をすすめてゆきたい

  • 代数多様体の射影空間への埋め込みに関する研究

    　View Summary

    Mを正標数の代数閉体上定義されている非特異完備代数多様体、i:M→Pを、像とbirationalとなるようなMから射影空間Pへの射とするとき、Mのgeneralな点xに対してixでのiMのembedded tangent spaceを対応させることにより得られるMからグラスマン多様体への ratio-nal mapを、(M,i)のガウス射という。平成5年度の研究目的は、ガウス者がほとんどの点で単射(generically injective)となるための、(M,i)に対する条件を明らかにすることであった。これについて、研究代表者は次を証明した。ただし、OMEGA:=Im(i^*:OMEGA^1_M←i^*OMEGA^1_P)とする:定理1:OMEGAがlocally free、ガウス射がfinite、さらに、OMEGAがgenerically ampleであれば、ガウス射はほとんどの点で単射である。 □さらに、早稲田大学理工学部助手、野間 淳氏との共同研究の成果として、定理1を改良しつつ、次を得た:定理2:種数2以上の非特異曲線C_iの直積C_1×・・・×C_mの非特異閉部分多様体は、どのように射影空間内に埋め込んでも、そのガウス射はほとんどの点で単射である。 □定理3:アーベル多様体の非特異閉部分多様体は、その法束が豊富ならば、どのように射影空間内に埋め込んでも、そのガウス射はほとんどの点で単射である。 □なお、以上の研究成果は、野間氏との共著論文“On the generic injectivity of the Gauss map in positive chatacteristic"にまとめられ、雑誌に投稿中である

  • An application of real aualytical wethad to nonlinear evolution eguations

    　View Summary

    We studied the initial-boundary value problem for Navier-Stokes equation in 3 dimensional exterior domain which describes the motion of incompressible viscous fluids. In 1930, J. Leray proved the existence of its weak solutions without uniqueness. But, we can not get any qualitative property of the motion of fluid from Leray's solutions. In 1950, R. Finn started to study the qualitative property of soltuions to the stationary Navier-Stokes equation, which was termed physically reasonable solutions (prs) by him. He proved the unique existence theorem of prs for small external force and the velocity uィイD2∞ィエD2 at infinity. After Finn, Heywood proved the stability of prs in the LィイD22ィエD2 framework, which now become one of the most important base of numerical investigation of the fluid motion. But, prs does not belong to LィイD22ィエD2 space, and therefore we have to study the stability of prs in the class to which prs belongs. This problem remained more than 30 years. In our present study, we solved this problem. We used the following argument. We took the Oseen approcimation in the 3 dim. Exterior domain, and then we investigated the local energy estimate near the boundary of optimal order of the Oseen equation by showing the fractional differentiablity of Oseen resolvent near the origin. Combinig this and LィイD2pィエD2-LィイD2qィエD2 estimate in the whole space by cut-off technique, we proved the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of the Oseen semigroup in 3 dim. Exterior domain. The most important point is that all the constant appearing in the estimate is independent of uィイD2∞ィエD2. By using this estimate, we could solve the stability problem of Finn's prs.(2) In 2 dimensinal case, we know the unique existence of Leray's weak solutions. But, since the Stokes fundamental solution has logarihmic singularity, we know less property of solution to NS in the exterior domain or even the whole space compared with 3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday, and therefore we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate (1< q≦ p≦ ∞) of Stokes semigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applying this estimate, we could show the best convergent rate of weak solutions to NS when time goes to infinity.(3) We considered the motion of the compressible viscous fluid too. Extending the method developed in study (1), we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain, which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity

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Syllabus

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Overseas Activities

  • 等質射影多様体の射影代数幾何的研究

    2000.09
    -
    2001.08

    ブラジル   ＩＭＰＡ

Sub-affiliation

  • Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

Research Institute

  • 2024
    -
    2026

    Waseda Research Institute for Science and Engineering   Concurrent Researcher

Internal Special Research Projects

  • 等質射影多様体のＬｉｅ環論的、表現論的、および、代数幾何的研究

    2006  

    　View Summary

    福井大学の保倉理美氏との共同研究により, ある種のreductive代数群の自然な作用に関する軌道分解について研究を行なった. 詳しくは, n次複素回転群とm次一般線型群の直積, SO(n) x GL(m) の, n行m列複素行列空間 M(n,m) への自然な作用による軌道分解を調べた. この概均質ベクトル空間に関する軌道分解は, すでに, 佐藤幹夫氏, 柏原正樹氏, 木村達雄氏, 大島利雄氏による著名な論文(M.Sato, M.Kashiwara, T.Kimura, T.Oshima: Micro-local analysis of prehomogeneous vector spaces, Invent. Math. 62 (1980), 117-179) において一例として取上げられているが, 保倉氏との共同研究においてそこに誤りがあることを発見した. 上記の我々の成果はそれを修正し完全な軌道分解を与えたものである. 一方, 射影多様体のreflexivityとガウス写像の分離性との関係については, これも長年研究を続けているテーマであるが, 未解決のまま残っていた2次元の場合にその同値性を証明することができた. これは広島大学の深澤知氏との共同研究の成果である. 従来の自分の研究成果と深澤氏の研究成果と合わせると, 1次元と2次元の場合は同値であり, 3次元以上の場合はreflexiveならガウス写像は分離的であるが, 逆は成り立たず, 任意正標数, 任意次元(≧3)の反例が存在することが示された. したがって, 任意標数, 任意次元においてreflexivityとガウス写像の分離性との関係が明らかになったことになる (標数零の場合は同値であることが古くから知られている).

  • 等質射影多様体の Lie 環論的, および, 代数幾何的研究

    1999  

    　View Summary

    　この数年間、等質射影多様体、特に、随伴多様体 (すなわち、複素単純代数群の随伴表現から得られる等質射影多様体) の研究を行っている。これは、H. Freudenthal の提唱するところの meta-symplectic geometry に相当する多様体であるが、symplectic geometry に相当する多様体として現れる等質射影多様体は、フロイデンタール多様体と呼ばれている。随伴多様体の研究を進めるにあたり、このフロイデンタール多様体を調べることが重要であることが、最近の研究から明らかになってきた。実際、1998年度の研究により、Lie環の接触型次数分解の次数1部分の定める部分空間と随伴多様体の交わりとしてフロイデンタール多様体が現れ、また、随伴多様体の secant variety の軌道分解を与える際にもこのフロイデンタール多様体、および、それを含む射影空間の考察が非常に重要であった。　そこで、1999年度は、このフロイデンタール多様体について詳しく研究を行った。その結果、フロイデンタール多様体の持ついくつかの顕著な射影幾何的性質を、symplectic triple system の理論の言葉で記述・証明することに成功した。これは、福井大学工学部、保倉理美氏との共同研究の成果である。　一方、射影幾何学的に重要な多様体のクラスのひとつに projective varieties with one apparent double point がある。1999年度は、多様体が等質の場合に、このクラスの多様体の分類を与えた。

  • 射影多様体の代数幾何的研究

    1997  

    　View Summary

    今年度は射影多様体の secant variety に関する研究を行った。特に射影多様体が随伴多様体 （すなわち、複素単純代数群の随伴表現から得られる等質射影多様体）の場合を中心に研究した。 　この場合、代数群の射影空間への作用は線型であることから直ちに、secant varietyも同じ代数群の作用を許すことがわかる。では、どのような軌道からsecant varietyが構成されているかが自然に問題となる。随伴多様体自身は明らかにその軌道のひとつである。昨年度は、secant variety の中で稠密となる軌道について明らかにした。今年度はそれら以外にも第3の軌道が存在することを、symplectic triple systems の理論 (浅野 洋氏; 横浜市立大学) を使うことにより示すことに成功した。　secant vareity の一般の点に対して、 その点を通る随伴多様体の射影空間に埋め込まれた接空間を考える。 その接点の全体のなす集合は随伴多様体の射影幾何学という立場からすると非常に興味深いが、今年度はその集合についての Lie 環論的特徴づけを発見した。 そこでは、 Lie 環に接触型次数構造を考えることが本質的であることがわかった。 　以上の結果の証明において、 ある等質空間を考えることが重要となる。 その空間は、 H. Freudenthal のsymplectic geometry に対応する：因みに、 ここで中心に考えている随伴多様体は彼の meta-symplectic geometry に対応し、一方、secant varietyの著しく退化した射影多様体のクラスとして現れる Severi 多様体は彼の projective geometry に対応している。今年度は、その等質空間と随伴多様体との関連を明らかにした。実は、随伴多様体の、ある線型部分空間による断面としてその等質空間が現れることがわかった。　以上は、福井大学工学部、保倉理美氏との共同研究の成果であり、現在論文にまとめている最中である。 研究成果の発表： 1. Adjoint varieties and their secant varieties, Indag. Math. (to appear) 2. Secant varieties of adjoint varieties, Matem&#193;　tica Contempor&#194;　nea (to appear) 3. Homogeneous projective varieties with degenerate secants, Trans. Amer. Math. Soc. (to appear)

  • 射影多様体の代数幾何学研究

    1996  

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    　一般に、射影多様体と整数d＞１に対して、多様体上のd+1点の張るd次元射影部分空間すべての和集合を考え、その閉包で定まる新しい射影多様体をd-secant varietyという。それに付随して、その退化の度合をはかる不変量、d-secant deficiencyが定まる。　研究代表者は、射影多様体が等質である場合、すなわち、代数群の有限次元既約表現から得られる場合のsecant varietyについて研究しており、昨年度はd=1の場合にsecant varietyの退化する射影多様体の分類を行ったが、今年度は主に、一般のduの場合にd-secant deficiencyの値について考察した。パソコンを用いるなどして実験的計算を行い、それを基にして、Al型の場合にいくつかのd-secant deficiencyに関する定理を得た。(計算データを基に一般の場合の命題を類推し、それに証明を与えた)。具体的には、射影空間のVeronese埋込、そして、Grassmann多様体のPlucker埋込について、d-secant deficiencyに関する公式を発見した。この結果は現在論文にまとめている最中である。　また、理工学部56号館において3月5日から8日まで、代数幾何学シンポジウム『射影多様体/代数多様体の射影幾何＋特異点』(理工総研、数理科学：研究代表者、足立恒雄からの援助による)を開催した。代数幾何学者だけでなく、対称空間、Lie環、非結合的代数、可換的代数など、さまざまな分野の専門家が参加し、講演に対しては活発に質問コメントなどがなされ、非常に盛況だった。上記の研究成果はこのシンポジウムで講演発表した。

  • 代数多様体の射影空間への埋め込みに関する研究

    1995