Basic algebraic geometry: 1 [Varieties in projective space]
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1994
|
| Edition: | 2., rev. and expanded ed. |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 303 S. |
| ISBN: | 9783540548126 3540548122 0387548122 |
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MARC
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| 020 | |a 9783540548126 |9 978-3-540-54812-6 | ||
| 020 | |a 3540548122 |9 3-540-54812-2 | ||
| 020 | |a 0387548122 |9 0-387-54812-2 | ||
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| 084 | |a SK 240 |0 (DE-625)143226: |2 rvk | ||
| 100 | 1 | |a Šafarevič, Igorʹ R. |d 1923-2017 |e Verfasser |0 (DE-588)119280337 |4 aut | |
| 240 | 1 | 0 | |a Osnovj algebraičeskoj geometrii |
| 245 | 1 | 0 | |a Basic algebraic geometry |n 1 |p [Varieties in projective space] |c Igor R. Shafarevich |
| 250 | |a 2., rev. and expanded ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1994 | |
| 300 | |a XVIII, 303 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 773 | 0 | 8 | |w (DE-604)BV009785411 |g 1 |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-006473381 | ||
Record in the Search Index
| _version_ | 1804124133185814528 |
|---|---|
| adam_text | Table
of
Contents Volume
1
BOOK
1.
Varieties in
Projective
Space
Chapter I. Basic Notions
................................... 1
1.
Algebraic Curves in the Plane
............................ 1
1.1.
Plane Curves
..................................... 1
1.2.
Rational Curves
.................................. 4
1.3.
Relation with Field Theory
......................... 8
1.4.
Rational Maps
.................................... 10
1.5.
Singular and Nonsingular Points
.................... 12
1.6.
The
Projective
Plane
.............................. 16
Exercises to
§1 ............................................ 21
2.
Closed Subsets of
Affine
Space
............................ 22
2.1.
Definition of Closed Subsets
........................ 22
2.2.
Regular Functions on a Closed Subset
............... 24
2.3.
Regular Maps
.................................... 27
Exercises to
§2 ............................................ 32
3.
Rational Functions
...................................... 34
3.1.
Irreducible Algebraic Subsets
....................... 34
3.2.
Rational Functions
................................ 35
3.3.
Rational Maps
.................................... 37
Exercises to
§3 ............................................ 40
4.
Quasiprojective Varieties
................................. 41
4.1.
Closed Subsets of
Projective
Space
.................. 41
4.2.
Regular Functions
................................ 46
4.3.
Rational Functions
................................ 50
4.4.
Examples of Regular Maps
......................... 52
Exercises to
§4 ............................................ 53
5.
Products and Maps of Quasiprojective Varieties
............. 54
5.1.
Products
......................................... 54
5.2.
The Image of
a Projective
Variety is Cloeed
.......... 57
5.3.
Finite Maps
...................................... 61
5.4.
Noether Normalisation
............................ 65
Exercises to
§5 ............................................ 66
6.
Dimension
............................................. 67
XII Table of
Contents
Volume
1
6.1. Definition
of
Dimension ............................ 67
6.2. Dimension
of Intersection with a Hypersurface
........ 70
6.3.
The Theorem on the Dimension of Fibres
............ 76
6.4.
Lines on Surfaces
................................. 78
Exercises to
§6 ............................................ 81
Chapter II. Local Properties
................................ 83
1.
Singular and Nonsingular Points
.......................... 83
1.1.
The Local Ring of a Point
......................... 83
1.2.
The Tangent Space
................................ 85
1.3.
Intrinsic Nature of the Tangent Space
............... 86
1.4.
Singular Points
................................... 92
1.5.
The Tangent Cone
................................ 95
Exercises to
§1 ............................................ 96
2.
Power Series Expansions
................................. 98
2.1.
Local Parameters at a Point
........................ 98
2.2.
Power Series Expansions
........................... 101
2.3.
Varieties over the Heals and the Complexes
.......... 104
Exercises to
§2 ............................................ 106
3.
Properties of Nonsingular Points
.......................... 107
3.1.
Codimension
1
Subvarieties
........................ 107
3.2.
Nonsingular Subvarieties
...........................
Ill
Exercises to
§3 ............................................ 112
4.
The Structure of
Birational
Maps
......................... 114
4.1.
Blowup in Protective Space
........................ 114
4.2.
Local Blowup
.................................... 115
4.3.
Behaviour of a Subvariety under a Blowup
........... 118
4.4.
Exceptional Subvarieties
........................... 119
4.5.
Isomorphism and
Birational
Equivalence
............. 121
Exercises to
§4 ............................................ 124
5.
Normal Varieties
........................................ 125
5.1.
Normal Varieties
.................................. 125
5.2.
Normalisation of an
Affine
Variety
.................. 129
5.3.
Normalisation of a Curve
.......................... 131
5.4.
Projective
Embedding of Nonsingular Varieties
....... 136
Exercises to
§5 ............................................ 138
6.
Singularities of a Map
................................... 139
6.1.
Irreducibility
..................................... 139
6.2.
Nonsingularity
.................................... 141
6.3.
Ramification
..................................... 142
6.4.
Examples
........................................ 146
Exercises to
§6 ............................................ 148
Table
of Contents Volume
1 XIII
Chapter III. Divisors and Differential Forms
................ 151
1.
Divisors
................................................ 151
1.1.
The Divisor of a Function
.......................... 151
1.2.
Locally Principal Divisors
.......................... 155
1.3.
Moving the Support of a Divisor away from a Point
... 158
1.4.
Divisors and Rational Maps
........................ 159
1.5.
The Linear System of a Divisor
..................... 161
1.6.
Pencil of Conies over P1
........................... 164
Exercises to
§1 ............................................ 166
2.
Divisors on Curves
...................................... 168
2.1.
The Degree of a Divisor on a Curve
................. 168
2.2.
Bezout s Theorem on a Curve
...................... 171
2.3.
The Dimension of a Divisor
......:................. 173
Exercises to
§2 ............................................ 174
3.
The Plane Cubic
........................................ 175
3.1.
The Class Group
.................................. 175
3.2.
The Group Law
.................................. 177
3.3.
Maps
............................................ 182
3.4.
Applications
...................................... 184
3.5.
Algebraically Nonclosed Field
...................... 185
Exercises to
§3 ............................................ 188
4.
Algebraic Groups
....................................... 188
4.1.
Algebraic Groups
................................. 188
4.2.
Quotient Groups and Chevalley s Theorem
........... 190
4.3.
Abelian Varieties
................................. 191
4.4.
The
Picard
Variety
................................ 192
Exercises to
§4 ............................................ 194
5.
Differential Forms
....................................... 195
5.1.
Regular Differential 1-forms
........................ 195
5.2.
Algebraic Definition of the Module of Differentials
----- 198
5.3.
Differential p-forms
................................ 199
5.4.
Rational Differential Forms
......................... 202
Exercises to
§5 ............................................ 204
6.
Examples and Applications of Differential Forms
............ 205
6.1.
Behaviour Under Maps
............................ 205
6.2.
Invariant Differential Forms on a Group
............. 207
6.3.
The Canonical Class
.............................. 209
6.4.
Hypersurfaces
.................................... 210
6.5.
Hyperelliptic Curves
.............................. 214
6.6.
The Riemann-Roch Theorem for Curves
............. 215
6.7.
Projective
Embedding of a Surface
.................. 218
Exercises to
§6 ............................................ 220
XIV Table of
Contents
Volume
1
Chapter IV. Intersection Numbers
.......................... 223
1.
Definition and Basic Properties
........................... 223
1.1.
Definition of Intersection Number
................... 223
1.2.
Additivity
....................................... 227
1.3.
Invariance
Under Linear Equivalence
................ 228
1.4.
The General Definition of Intersection Number
....... 232
Exercises to
§1 ............................................ 235
2.
Applications of Intersection Numbers
...................... 236
2.1.
Bézouťs
Theorem in
Projective
and Multiprojective
Space
............................................... 236
2.2.
Varieties over the Reals
............................ 238
2.3.
The Genus of a Nonsingular Curve on a Surface
...... 241
2.4.
The Riemann-Roch Inequality on a Surface
.......... 244
2.5.
The Nonsingular Cubic Surface
..................... 246
2.6.
The Ring of Cycle Classes
.......................... 249
Exercises to
§2 ............................................ 250
3. Birational
Maps of Surfaces
___.......................... 251
3.1.
Blowups of Surfaces
............................... 251
3.2.
Some Intersection Numbers
........................ 252
3.3.
Resolution of Indeterminacy
........................ 254
3.4.
Factorisation as a Chain of Blowups
................. 256
3.5.
Remarks and Examples
............................ 258
Exercises to
§3 ............................................ 260
4.
Singularities
............................................ 261
4.1.
Singular Points of a Curve
......................... 261
4.2.
Surface Singularities
............................... 264
4.3. Du
Val
Singularities
............................... 266
4.4.
Degeneration of Curves
............................ 270
Exercises to
§4 ............................................ 273
Algebraic Appendix
........................................ 275
1.
Linear and Bilinear Algebra
......................... 275
2.
Polynomials
....................................... 277
3. Quasilinear
Maps
................................... 277
4.
Invariants
......................................... 279
5.
Fields
............................................. 280
6.
Commutative Rings
................................. 281
7.
Unique Factorisation
................................ 284
8.
Integral Elements
................................... 286
9.
Length of a Module
................................. 286
References
.................................................. 289
Index
....................................................... 293
|
| any_adam_object | 1 |
| author | Šafarevič, Igorʹ R. 1923-2017 |
| author_GND | (DE-588)119280337 |
| author_facet | Šafarevič, Igorʹ R. 1923-2017 |
| author_role | aut |
| author_sort | Šafarevič, Igorʹ R. 1923-2017 |
| author_variant | i r š ir irš |
| building | Verbundindex |
| bvnumber | BV009785417 |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)243794419 (DE-599)BVBBV009785417 |
| discipline | Mathematik |
| edition | 2., rev. and expanded ed. |
| format | Book |
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| id | DE-604.BV009785417 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:40:49Z |
| institution | BVB |
| isbn | 9783540548126 3540548122 0387548122 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006473381 |
| oclc_num | 243794419 |
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| owner | DE-355 DE-BY-UBR DE-29T DE-384 DE-739 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-703 DE-12 DE-20 DE-19 DE-BY-UBM DE-634 DE-83 DE-11 DE-188 |
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| physical | XVIII, 303 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Springer |
| record_format | marc |
| spelling | Šafarevič, Igorʹ R. 1923-2017 Verfasser (DE-588)119280337 aut Osnovj algebraičeskoj geometrii Basic algebraic geometry 1 [Varieties in projective space] Igor R. Shafarevich 2., rev. and expanded ed. Berlin [u.a.] Springer 1994 XVIII, 303 S. txt rdacontent n rdamedia nc rdacarrier (DE-604)BV009785411 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006473381&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Šafarevič, Igorʹ R. 1923-2017 Basic algebraic geometry |
| title | Basic algebraic geometry |
| title_alt | Osnovj algebraičeskoj geometrii |
| title_auth | Basic algebraic geometry |
| title_exact_search | Basic algebraic geometry |
| title_full | Basic algebraic geometry 1 [Varieties in projective space] Igor R. Shafarevich |
| title_fullStr | Basic algebraic geometry 1 [Varieties in projective space] Igor R. Shafarevich |
| title_full_unstemmed | Basic algebraic geometry 1 [Varieties in projective space] Igor R. Shafarevich |
| title_short | Basic algebraic geometry |
| title_sort | basic algebraic geometry varieties in projective space |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006473381&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009785411 |
| work_keys_str_mv | AT safarevicigorʹr osnovjalgebraiceskojgeometrii AT safarevicigorʹr basicalgebraicgeometry1 |