Ruled varieties: an introduction to algebraic differential geometry
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Braunschweig [u.a.]
Vieweg
2001
|
| Edition: | 1. ed. |
| Series: | Advanced lectures in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 141 S. |
| ISBN: | 3528031387 |
Staff View
MARC
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| 245 | 1 | 0 | |a Ruled varieties |b an introduction to algebraic differential geometry |c Gerd Fischer ; Jens Piontkowski |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Braunschweig [u.a.] |b Vieweg |c 2001 | |
| 300 | |a X, 141 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Advanced lectures in mathematics | |
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| 650 | 4 | |a Géométrie algébrique | |
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Record in the Search Index
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| adam_text | Contents
0 Review from Classical Differential and Projective Geometry 1
0.1 Developable Rulings 1
0.2 Vanishing GauB Curvature 3
0.3 Hessian Matrices 5
0.4 Classification of Developable Surfaces in K3 7
0.5 Developable Surfaces in P3(C) 9
1 Grassmannians 12
1.1 Preliminaries 12
1.1.1 Algebraic Varieties 12
1.1.2 Rational Maps 16
1.1.3 Holomorphic Linear Combinations 19
1.1.4 Limit Direction of a Holomorphic Path 20
1.1.5 Radial Paths 21
1.2 Plucker Coordinates 22
1.2.1 Local Coordinates 22
1.2.2 The Plucker Embedding 23
1.2.3 Lines in P3 24
1.2.4 The Plucker Image 25
1.2.5 Plucker Relations 27
1.2.6 Systems of Vector Valued Functions 29
1.3 Incidences and Duality 31
1.3.1 Equations and Generators in Terms of Plucker Coordinates 31
1.3.2 Flag Varieties 32
1.3.3 Duality of Grassmannians 33
1.3.4 Dual Projective Spaces 33
VIII
1.4 Tangents to Grassmannians 34
1.4.1 Tangents to Projective Space 34
1.4.2 The Tangent Space of the Grassmannian 35
1.5 Curves in Grassmannians 37
1.5.1 The Drill 37
1.5.2 Derived Curves 39
1.5.3 Sums and Intersections 43
1.5.4 Associated Curves and Curves with Prescribed Drill 45
1.5.5 Normal Form 47
2 Ruled Varieties 49
2.1 Incidence Varieties and Duality 49
2.1.1 Unions of Linear Varieties 49
2.1.2 Fano Varieties 50
2.1.3 Joins 51
2.1.4 Conormal Bundle and Dual Variety 52
2.1.5 Duality Theorem 55
2.1.6 The Contact Locus 56
2.1.7 The Dual Curve 57
2.1.8 Rational Curves 59
2.2 Developable Varieties 61
2.2.1 Rulings 61
2.2.2 Adapted Parameterizations 63
2.2.3 Germs of Rulings 64
2.2.4 Developable Rulings and Focal Points 65
2.2.5 Developability of Joins 69
2.2.6 Dual Varieties of Cones and Degenerate Varieties 71
2.2.7 Tangent and Osculating Scrolls 74
2.2.8 Classification of Developable One Parameter Rulings 77
_IX
2.2.9 Example of a Twisted Plane 78
2.2.10 Characterization of Drill One Curves 82
2.3 TheGauBMap 85
2.3.1 Definition of the GauB Map 85
2.3.2 Linearity of the Fibers 86
2.3.3 GauB Map and Developability 88
2.3.4 GauB Image and Dual Variety 88
2.3.5 Existence of Varieties with Given GauB Rank 89
2.4 The Second Fundamental Form 93
2.4.1 Definition of the Second Fundamental Form 93
2.4.2 The Degeneracy Space 96
2.4.3 The Degeneracy Map 97
2.4.4 The Singular and Base Locus 98
2.4.5 The Codimension of a Uniruled Variety 99
2.4.6 Fibers of the GauB Map 101
2.4.7 Characterization of GauB Images 103
2.4.8 Singularities of the GauB Map 106
2.5 GauB Defect and Dual Defect 109
2.5.1 Dual Defect of Segre Varieties 110
2.5.2 GauB Defect and Singular Locus Ill
2.5.3 Dual Defect and Singular Locus 112
2.5.4 Computation of the Dual Defect 113
2.5.5 The Surface Case 115
2.5.6 Classification of Developable Hypersurfaces 116
2.5.7 Dual Defect of Uniruled Varieties 117
2.5.8 Varieties with Very Small Dual Varieties 118
X
3 Tangent and Secant Varieties 119
3.1 Zak s Theorems 119
3.1.1 Tangent Spaces, Tangent Cones, and Tangent Stars 119
3.1.2 Zak s Theorem on Tangent and Secant Varieties 121
3.1.3 Theorem on Tangencies 124
3.2 Third and Higher Fundamental Forms 125
3.2.1 Definition 125
3.2.2 Vanishing of Fundamental Forms 128
3.3 Tangent Varieties 129
3.3.1 The Dimension of the Tangent Variety 129
3.3.2 Developability of the Tangent Variety 130
3.3.3 Singularities of the Tangent Variety 133
3.4 The Dimension of the Secant Variety 135
Bibliography 137
Index 140
List of Symbols 142
|
| any_adam_object | 1 |
| author | Fischer, Gerd 1939- Piontkowski, Jens |
| author_GND | (DE-588)128621052 |
| author_facet | Fischer, Gerd 1939- Piontkowski, Jens |
| author_role | aut aut |
| author_sort | Fischer, Gerd 1939- |
| author_variant | g f gf j p jp |
| building | Verbundindex |
| bvnumber | BV013728540 |
| callnumber-first | Q - Science |
| callnumber-label | QA184 |
| callnumber-raw | QA184 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| classification_tum | MAT 140f MAT 530f |
| ctrlnum | (OCoLC)48399774 (DE-599)BVBBV013728540 |
| dewey-full | 512.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.5 |
| dewey-search | 512.5 |
| dewey-sort | 3512.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| indexdate | 2024-07-09T18:50:57Z |
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| isbn | 3528031387 |
| language | English |
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| series2 | Advanced lectures in mathematics |
| spelling | Fischer, Gerd 1939- Verfasser (DE-588)128621052 aut Ruled varieties an introduction to algebraic differential geometry Gerd Fischer ; Jens Piontkowski 1. ed. Braunschweig [u.a.] Vieweg 2001 X, 141 S. txt rdacontent n rdamedia nc rdacarrier Advanced lectures in mathematics Algèbre linéaire Géométrie algébrique Algebras, Linear Geometry, Algebraic Tangentenvarietät (DE-588)4640697-9 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Abwickelbare Varietät (DE-588)4640698-0 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Geregelte Varietät (DE-588)4640695-5 gnd rswk-swf Sekantenvarietät (DE-588)4640696-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 Geregelte Varietät (DE-588)4640695-5 s Abwickelbare Varietät (DE-588)4640698-0 s Sekantenvarietät (DE-588)4640696-7 s Tangentenvarietät (DE-588)4640697-9 s Piontkowski, Jens Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009382852&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fischer, Gerd 1939- Piontkowski, Jens Ruled varieties an introduction to algebraic differential geometry Algèbre linéaire Géométrie algébrique Algebras, Linear Geometry, Algebraic Tangentenvarietät (DE-588)4640697-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Abwickelbare Varietät (DE-588)4640698-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geregelte Varietät (DE-588)4640695-5 gnd Sekantenvarietät (DE-588)4640696-7 gnd |
| subject_GND | (DE-588)4640697-9 (DE-588)4001161-6 (DE-588)4640698-0 (DE-588)4012248-7 (DE-588)4640695-5 (DE-588)4640696-7 |
| title | Ruled varieties an introduction to algebraic differential geometry |
| title_auth | Ruled varieties an introduction to algebraic differential geometry |
| title_exact_search | Ruled varieties an introduction to algebraic differential geometry |
| title_full | Ruled varieties an introduction to algebraic differential geometry Gerd Fischer ; Jens Piontkowski |
| title_fullStr | Ruled varieties an introduction to algebraic differential geometry Gerd Fischer ; Jens Piontkowski |
| title_full_unstemmed | Ruled varieties an introduction to algebraic differential geometry Gerd Fischer ; Jens Piontkowski |
| title_short | Ruled varieties |
| title_sort | ruled varieties an introduction to algebraic differential geometry |
| title_sub | an introduction to algebraic differential geometry |
| topic | Algèbre linéaire Géométrie algébrique Algebras, Linear Geometry, Algebraic Tangentenvarietät (DE-588)4640697-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Abwickelbare Varietät (DE-588)4640698-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geregelte Varietät (DE-588)4640695-5 gnd Sekantenvarietät (DE-588)4640696-7 gnd |
| topic_facet | Algèbre linéaire Géométrie algébrique Algebras, Linear Geometry, Algebraic Tangentenvarietät Algebraische Geometrie Abwickelbare Varietät Differentialgeometrie Geregelte Varietät Sekantenvarietät |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009382852&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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