A concise introduction to algebraic varieties:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
AMS American Mathematical Society
[2021]
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| Series: | Graduate studies in mathematics
216 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index 2109 |
| Physical Description: | xvi, 259 Seiten Illustrationen |
| ISBN: | 9781470460136 9781470466657 |
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MARC
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Record in the Search Index
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| adam_text | Contents Preface Chapter 1. xi Introduction: An overview of algebraic geometry through the lens of plane curves 1 §1.1 . Plane curves 1 §1.2 . Elliptic curves 5 Chapter 2. Affine algebraic varieties 11 §2.1 . Zero sets and the Zariski topology 11 §2.2 . Zero sets and ideals 15 §2.3 . Noetherian spaces 22 §2.4 . Dimension 24 Chapter 3. Regular functions and morphisms 31 §3.1 . Regular functions 31 §3.2 . Morphisms 39 §3.3 . Rational maps 48 §3.4 . Chevalley’s theorem 52 §3.A . Recovering geometry from categories 55 Chapter 4. Singularities 59 §4.1 . Tangent lines and singularities 60 §4.2 . Zariski cotangent spaces 62 §4.3 . The Jacobian criterion 65 §4.4 . Completions and power series 71 vii
Contents viii §4-5. Normality and normalization 74 §4.A. Local generation of ideals 77 Chapter 5. Abstract varieties via atlases §5.1. Prevarieties 81 81 §5.2. §5.3. Regular functions and morphisms 86 Abstract varieties Normalization revisited 90 §5.4. Chapter 6. §6.1. Projective varieties Projective space §6.2. Projective varieties and morphisms §6.3. Blowup of subvarieties §6.A. Homogeneous ideals and coordinate rings Chapter 7. Nonsingular curves and complete varieties §7.1. Curves, regular functions, and morphisms §7.2. Quasiprojectivity §7.3. §7.4. 96 103 103 108 114 122 125 125 130 §7.5. Projective curves Completeness A limit-based criterion 132 135 137 §7.6. Irreducibility of polynomials in families 141 Chapter 8. §8.1. §8-2. §8.3. §8.4. §8.5. Morphisms of curves Divisors on curves Linear equivalence and morphisms to projective space Embeddings of curves Secant varieties and curves in projective space Chapter 9. §9.1. Divisors on nonsingular curves Differential forms Differential forms §9.2. Differential forms on curves §9.3. Differential forms and ramification §9.A. Field automorphisms and Frobenius Chapter 10. §10.1. §10.2. An invitation to the theory of algebraic curves The Riemann-Roch theorem The Riemann-Hurwitz theorem 147 147 150 153 160 163 169 169 173 176 181 187 187 190
Contents ix §10.3 . Brill-Noether theory and moduli spaces of curves §10.A . Remarks on proofs of the Riemann-Roch theorem 195 199 Appendix A. Complex varieties and the analytic topology §A.l. Quasiaffine complex varieties §A.2. The analytic topology on prevarieties §A.3. Fundamental results §A.4. Nonsingularity and complex manifolds §A.5. Connectedness 211 212 213 Appendix B. 217 A roadmap through algebra Field theory §B.l . 207 207 209 218 §B.10 . Noether normalization and firstapplications §B.ll . Dimension theory over fields §B.12 . Extensions of Dedekind domains 221 222 223 225 228 229 231 232 236 238 241 §B.13 . Completion and power series 244 §B.2 . Algebras §B.3 . Noetherian rings §B.4 . Rings of fractions §B.5 §B.6 §B.7 . . . Nakayama’s lemma Unique factorization Integral extensions §B.8 §B.9 . . Integral closure The principal ideal theorem and regular local rings Bibliography 247 Index of Notation 251 Index 253
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| adam_txt |
Contents Preface Chapter 1. xi Introduction: An overview of algebraic geometry through the lens of plane curves 1 §1.1 . Plane curves 1 §1.2 . Elliptic curves 5 Chapter 2. Affine algebraic varieties 11 §2.1 . Zero sets and the Zariski topology 11 §2.2 . Zero sets and ideals 15 §2.3 . Noetherian spaces 22 §2.4 . Dimension 24 Chapter 3. Regular functions and morphisms 31 §3.1 . Regular functions 31 §3.2 . Morphisms 39 §3.3 . Rational maps 48 §3.4 . Chevalley’s theorem 52 §3.A . Recovering geometry from categories 55 Chapter 4. Singularities 59 §4.1 . Tangent lines and singularities 60 §4.2 . Zariski cotangent spaces 62 §4.3 . The Jacobian criterion 65 §4.4 . Completions and power series 71 vii
Contents viii §4-5. Normality and normalization 74 §4.A. Local generation of ideals 77 Chapter 5. Abstract varieties via atlases §5.1. Prevarieties 81 81 §5.2. §5.3. Regular functions and morphisms 86 Abstract varieties Normalization revisited 90 §5.4. Chapter 6. §6.1. Projective varieties Projective space §6.2. Projective varieties and morphisms §6.3. Blowup of subvarieties §6.A. Homogeneous ideals and coordinate rings Chapter 7. Nonsingular curves and complete varieties §7.1. Curves, regular functions, and morphisms §7.2. Quasiprojectivity §7.3. §7.4. 96 103 103 108 114 122 125 125 130 §7.5. Projective curves Completeness A limit-based criterion 132 135 137 §7.6. Irreducibility of polynomials in families 141 Chapter 8. §8.1. §8-2. §8.3. §8.4. §8.5. Morphisms of curves Divisors on curves Linear equivalence and morphisms to projective space Embeddings of curves Secant varieties and curves in projective space Chapter 9. §9.1. Divisors on nonsingular curves Differential forms Differential forms §9.2. Differential forms on curves §9.3. Differential forms and ramification §9.A. Field automorphisms and Frobenius Chapter 10. §10.1. §10.2. An invitation to the theory of algebraic curves The Riemann-Roch theorem The Riemann-Hurwitz theorem 147 147 150 153 160 163 169 169 173 176 181 187 187 190
Contents ix §10.3 . Brill-Noether theory and moduli spaces of curves §10.A . Remarks on proofs of the Riemann-Roch theorem 195 199 Appendix A. Complex varieties and the analytic topology §A.l. Quasiaffine complex varieties §A.2. The analytic topology on prevarieties §A.3. Fundamental results §A.4. Nonsingularity and complex manifolds §A.5. Connectedness 211 212 213 Appendix B. 217 A roadmap through algebra Field theory §B.l . ' 207 207 209 218 §B.10 . Noether normalization and firstapplications §B.ll . Dimension theory over fields §B.12 . Extensions of Dedekind domains 221 222 223 225 228 229 231 232 236 238 241 §B.13 . Completion and power series 244 §B.2 . Algebras §B.3 . Noetherian rings §B.4 . Rings of fractions §B.5 §B.6 §B.7 . . . Nakayama’s lemma Unique factorization Integral extensions §B.8 §B.9 . . Integral closure The principal ideal theorem and regular local rings Bibliography 247 Index of Notation 251 Index 253 |
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| index_date | 2024-07-03T18:51:49Z |
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| institution | BVB |
| isbn | 9781470460136 9781470466657 |
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| spelling | Osserman, Brian 1977- Verfasser (DE-588)1259271854 aut A concise introduction to algebraic varieties Brian Osserman Providence, Rhode Island AMS American Mathematical Society [2021] xvi, 259 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 216 Includes bibliographical references and index 2109 Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd rswk-swf Algebraic varieties Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.) Algebraic geometry -- Foundations -- Varieties and morphisms Algebraic geometry -- Local theory -- Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] Algebraische Varietät (DE-588)4581715-7 s Algebraische Mannigfaltigkeit (DE-588)4128509-8 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-6664-0 Graduate studies in mathematics 216 (DE-604)BV009739289 216 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033043375&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Osserman, Brian 1977- A concise introduction to algebraic varieties Graduate studies in mathematics Algebraische Varietät (DE-588)4581715-7 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd |
| subject_GND | (DE-588)4581715-7 (DE-588)4128509-8 |
| title | A concise introduction to algebraic varieties |
| title_auth | A concise introduction to algebraic varieties |
| title_exact_search | A concise introduction to algebraic varieties |
| title_exact_search_txtP | A concise introduction to algebraic varieties |
| title_full | A concise introduction to algebraic varieties Brian Osserman |
| title_fullStr | A concise introduction to algebraic varieties Brian Osserman |
| title_full_unstemmed | A concise introduction to algebraic varieties Brian Osserman |
| title_short | A concise introduction to algebraic varieties |
| title_sort | a concise introduction to algebraic varieties |
| topic | Algebraische Varietät (DE-588)4581715-7 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd |
| topic_facet | Algebraische Varietät Algebraische Mannigfaltigkeit |
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