A first course in algebraic geometry and algebraic varieties:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2023]
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| Series: | Essential textbooks in mathematics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xix, 305 Seiten Illustrationen, Diagramme |
| ISBN: | 9781800612747 |
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| 100 | 1 | |a Flamini, Flaminio |e Verfasser |0 (DE-588)1302433717 |4 aut | |
| 245 | 1 | 0 | |a A first course in algebraic geometry and algebraic varieties |c Flaminio Flamini, University of Rome "Tor Vergata", Italy |
| 246 | 1 | 3 | |a Algebraic geometry and algebraic varieties |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2023] | |
| 300 | |a xix, 305 Seiten |b Illustrationen, Diagramme | ||
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Contents Preface vii About the Author xi Acknowledgments xiii 1. Basics on Commutative Algebra Idealsand Operations on Ideals. UFDs and PIDs. Polynomial Rings. 1.3.1 Polynomials inՍիր], where D a UFD. 1.3.2 The case D = Ka field. 1.3.3 Resultant of two polynomials in D[a;] . 1.3.4 Resultant in D[a?i,., rn] and elimination . 1.4 Noetherian Rings and the Hilbert Basis Theorem . 1.5 Д-Modules, Д-Algebras and Finiteness Conditions . 1.6 Integrality. 19 1.7 Zariski’s Lemma. 21 1.8 Transcendence Degree. 23 1.9 Tensor Products of Д-Modules and of Д-Algebras . 1.9.1 Restriction and extension of scalars. 1.9.2 Tensor product of algebras. 1.10 Graded Rings and Modules, Homogeneous Ideals . 1.10.1 Homogeneous polynomials . 1.10.2 Graded modules and graded morphisms. 1.1 1.2 1.3 XV 1 2 4 5 5 8 9 12 13 16 27 29 30 31 35 41
xvi 2. 3. A Firet Course in Algebraic Geometry and Algebraic Varieties 1.11 Localization. 1.11.1 Local rings and localization. 1.12 Krull-Dimension of a Ring. Exercises. 42 46 48 52 Algebraic Affine Sets 55 2.1 Algebraic Affine Sets and Ideals. 2.2 Hilbert “Nullstellensatz”. 2.3 Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory. 2.3.1 Study’s principle. 2.3.2 Intersections of affine plane curves. Exercises. 55 70 74 74 75 76 Algebraic Projective Sets 79 Algebraic Projective Sets. 80 Homogeneous “Hilbert Nullstellensatz”. 83 Fundamental Examples and Remarks. 86 3.3.1 Points. 86 3.3.2 Coordinate linear subspaces. 87 3.3.3 Hyperplanes and the dual projective space . 87 3.3.4 Fundamental affine open sets (or affine charts) of Fn . 87 3.3.5 Projective
closure of affine sets. 89 3.3.6 Projective subspaces and their ideals. 91 3.3.7 Projective and affine subspaces. 94 3.3.8 Homographies, projectivities andaffinities . 95 3.3.9 Projective cones. 99 3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces. 99 3.3.11 Proper closed subsets of F2. 100 3.3.12 Affine and projective twisted cubics. 101 Exercises. 106 3.1 3.2 3.3 4. Topological Properties and Algebraic Varieties 4.1 109 Irreducible Topological Spaces. 109 4.1.1 Coordinate rings, ideals and irreducibility . 112
xvii Contente 4.1.2 Algebraic varieties. 114 4.2 Noetherian Spaces: Irreducible Components. 115 4.3 Combinatorial Dimension . 118 Exercises. 123 5. Regular and Rational Functions on Algebraic Varieties 125 Basics on Sheaves. 125 Regular Functions. 127 Rational Functions. 130 5.3.1 Consequences of the fundamental theorem on regular and rational functions. 141 5.3.2 Examples. 143 Exercises. 147 5.1 5.2 5.3 6. Morphisms of Algebraic Varieties 149 Morphisms. Morphisms with (Quasi) Affine Target. Morphisms with (Quasi) Projective Target. Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety. 6.5 Veronese Morphism: Divisors and Linear Systems . 6.5.1 Veronese morphism and consequences. 6.5.2 Divisors and linear systems.
Exercises. 149 151 160 164 166 171 174 177 Products of Algebraic Varieties 179 6.1 6.2 6.3 6.4 7. Products of Affine Varieties. Products of Projective Varieties. 7.2.1 Segre morphism and the product of projective spaces. 184 7.2.2 Products of projective varieties. 7.3 Products of Algebraic Varieties. 7.4 Products of Morphisms. 7.5 Diagonals, Graph of a Morphism and Fiber-Products. 191 Exercises. 7.1 7.2 179 182 186 188 189 193
xviii 8. A First Course in Algebraic Geometry and Algebraic Varieties Rational Maps of Algebraic Varieties 195 Rational and Birational Maps. 195 8.1.1 Some properties and some examples of (bi)rationai maps. 199 8.2 Unirational and Rational Varieties . 205 8.2.1 Stereographic projection of a rank-four quadric surface. 206 8.2.2 Monoids. 208 8.2.3 Blow-up of F1 at a point.209 8.2.4 Blow-ups and resolution ofsingularities. 212 Exercises. 215 8.1 9. Completeness of Projective Varieties 217 Complete Algebraic Varieties .217 The Main Theorem of EliminationTheory. 219 9.2.1 Consequences of the main theorem of elimination theory. 221 Exercises. 222 9.1 9.2 10. Dimension of Algebraic Varieties 10.1 Dimension of an Algebraic Variety . 10.2 Comparison on Various Definitions of “Dimension”. 10.3 Dimension and Intersections. 10.4 Complete
Intersections. Exercises. 11. Fiber-Dimension: Semicontinuity 11.1 Fibers of a Dominant Morphism. 11.2 Semicontinuity . Exercises. 12. Tangent Spaces: Smoothness of Algebraic Varieties 12.1 Tangent Space at a Point of an Affine Variety: Smoothness. 12.2 Tangent Space at a Point of a Projective Variety: Smoothness. 253 225 225 230 232 235 239 241 241 244 246 249 249
Contents 12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definitionof Smoothness. Exercises. xix 255 263 Solutions to Exercises 265 Bibliography 295 Index 297 |
| adam_txt |
Contents Preface vii About the Author xi Acknowledgments xiii 1. Basics on Commutative Algebra Idealsand Operations on Ideals. UFDs and PIDs. Polynomial Rings. 1.3.1 Polynomials inՍիր], where D a UFD. 1.3.2 The case D = Ka field. 1.3.3 Resultant of two polynomials in D[a;] . 1.3.4 Resultant in D[a?i,., rn] and elimination . 1.4 Noetherian Rings and the Hilbert Basis Theorem . 1.5 Д-Modules, Д-Algebras and Finiteness Conditions . 1.6 Integrality. 19 1.7 Zariski’s Lemma. 21 1.8 Transcendence Degree. 23 1.9 Tensor Products of Д-Modules and of Д-Algebras . 1.9.1 Restriction and extension of scalars. 1.9.2 Tensor product of algebras. 1.10 Graded Rings and Modules, Homogeneous Ideals . 1.10.1 Homogeneous polynomials . 1.10.2 Graded modules and graded morphisms. 1.1 1.2 1.3 XV 1 2 4 5 5 8 9 12 13 16 27 29 30 31 35 41
xvi 2. 3. A Firet Course in Algebraic Geometry and Algebraic Varieties 1.11 Localization. 1.11.1 Local rings and localization. 1.12 Krull-Dimension of a Ring. Exercises. 42 46 48 52 Algebraic Affine Sets 55 2.1 Algebraic Affine Sets and Ideals. 2.2 Hilbert “Nullstellensatz”. 2.3 Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory. 2.3.1 Study’s principle. 2.3.2 Intersections of affine plane curves. Exercises. 55 70 74 74 75 76 Algebraic Projective Sets 79 Algebraic Projective Sets. 80 Homogeneous “Hilbert Nullstellensatz”. 83 Fundamental Examples and Remarks. 86 3.3.1 Points. 86 3.3.2 Coordinate linear subspaces. 87 3.3.3 Hyperplanes and the dual projective space . 87 3.3.4 Fundamental affine open sets (or affine charts) of Fn . 87 3.3.5 Projective
closure of affine sets. 89 3.3.6 Projective subspaces and their ideals. 91 3.3.7 Projective and affine subspaces. 94 3.3.8 Homographies, projectivities andaffinities . 95 3.3.9 Projective cones. 99 3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces. 99 3.3.11 Proper closed subsets of F2. 100 3.3.12 Affine and projective twisted cubics. 101 Exercises. 106 3.1 3.2 3.3 4. Topological Properties and Algebraic Varieties 4.1 109 Irreducible Topological Spaces. 109 4.1.1 Coordinate rings, ideals and irreducibility . 112
xvii Contente 4.1.2 Algebraic varieties. 114 4.2 Noetherian Spaces: Irreducible Components. 115 4.3 Combinatorial Dimension . 118 Exercises. 123 5. Regular and Rational Functions on Algebraic Varieties 125 Basics on Sheaves. 125 Regular Functions. 127 Rational Functions. 130 5.3.1 Consequences of the fundamental theorem on regular and rational functions. 141 5.3.2 Examples. 143 Exercises. 147 5.1 5.2 5.3 6. Morphisms of Algebraic Varieties 149 Morphisms. Morphisms with (Quasi) Affine Target. Morphisms with (Quasi) Projective Target. Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety. 6.5 Veronese Morphism: Divisors and Linear Systems . 6.5.1 Veronese morphism and consequences. 6.5.2 Divisors and linear systems.
Exercises. 149 151 160 164 166 171 174 177 Products of Algebraic Varieties 179 6.1 6.2 6.3 6.4 7. Products of Affine Varieties. Products of Projective Varieties. 7.2.1 Segre morphism and the product of projective spaces. 184 7.2.2 Products of projective varieties. 7.3 Products of Algebraic Varieties. 7.4 Products of Morphisms. 7.5 Diagonals, Graph of a Morphism and Fiber-Products. 191 Exercises. 7.1 7.2 179 182 186 188 189 193
xviii 8. A First Course in Algebraic Geometry and Algebraic Varieties Rational Maps of Algebraic Varieties 195 Rational and Birational Maps. 195 8.1.1 Some properties and some examples of (bi)rationai maps. 199 8.2 Unirational and Rational Varieties . 205 8.2.1 Stereographic projection of a rank-four quadric surface. 206 8.2.2 Monoids. 208 8.2.3 Blow-up of F1 at a point.209 8.2.4 Blow-ups and resolution ofsingularities. 212 Exercises. 215 8.1 9. Completeness of Projective Varieties 217 Complete Algebraic Varieties .217 The Main Theorem of EliminationTheory. 219 9.2.1 Consequences of the main theorem of elimination theory. 221 Exercises. 222 9.1 9.2 10. Dimension of Algebraic Varieties 10.1 Dimension of an Algebraic Variety . 10.2 Comparison on Various Definitions of “Dimension”. 10.3 Dimension and Intersections. 10.4 Complete
Intersections. Exercises. 11. Fiber-Dimension: Semicontinuity 11.1 Fibers of a Dominant Morphism. 11.2 Semicontinuity . Exercises. 12. Tangent Spaces: Smoothness of Algebraic Varieties 12.1 Tangent Space at a Point of an Affine Variety: Smoothness. 12.2 Tangent Space at a Point of a Projective Variety: Smoothness. 253 225 225 230 232 235 239 241 241 244 246 249 249
Contents 12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definitionof Smoothness. Exercises. xix 255 263 Solutions to Exercises 265 Bibliography 295 Index 297 |
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| index_date | 2024-07-03T21:44:48Z |
| indexdate | 2024-07-20T05:49:49Z |
| institution | BVB |
| isbn | 9781800612747 |
| language | English |
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| spelling | Flamini, Flaminio Verfasser (DE-588)1302433717 aut A first course in algebraic geometry and algebraic varieties Flaminio Flamini, University of Rome "Tor Vergata", Italy Algebraic geometry and algebraic varieties New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2023] xix, 305 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Essential textbooks in mathematics Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Algebraische Varietät (DE-588)4581715-7 s DE-604 Erscheint auch als Online-Ausgabe 978-1-80061-266-2 Erscheint auch als Online-Ausgabe 978-1-80061-267-9 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034140022&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Flamini, Flaminio A first course in algebraic geometry and algebraic varieties Algebraische Varietät (DE-588)4581715-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4581715-7 (DE-588)4001161-6 |
| title | A first course in algebraic geometry and algebraic varieties |
| title_alt | Algebraic geometry and algebraic varieties |
| title_auth | A first course in algebraic geometry and algebraic varieties |
| title_exact_search | A first course in algebraic geometry and algebraic varieties |
| title_exact_search_txtP | A first course in algebraic geometry and algebraic varieties |
| title_full | A first course in algebraic geometry and algebraic varieties Flaminio Flamini, University of Rome "Tor Vergata", Italy |
| title_fullStr | A first course in algebraic geometry and algebraic varieties Flaminio Flamini, University of Rome "Tor Vergata", Italy |
| title_full_unstemmed | A first course in algebraic geometry and algebraic varieties Flaminio Flamini, University of Rome "Tor Vergata", Italy |
| title_short | A first course in algebraic geometry and algebraic varieties |
| title_sort | a first course in algebraic geometry and algebraic varieties |
| topic | Algebraische Varietät (DE-588)4581715-7 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Algebraische Varietät Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034140022&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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