Basic algebraic geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1977
|
| Edition: | Rev. print. |
| Series: | Grundlehren der mathematischen Wissenschaften
213 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 439 S. graph. Darst. |
| ISBN: | 3540066918 0387066918 |
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MARC
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| 001 | BV042181398 | ||
| 003 | DE-604 | ||
| 005 | 20250411 | ||
| 007 | t| | ||
| 008 | 141112s1977 xx d||| |||| 00||| eng d | ||
| 015 | |a 77A360877 |2 dnb | ||
| 016 | 7 | |a 770337228 |2 DE-101 | |
| 020 | |a 3540066918 |9 3-540-06691-8 | ||
| 020 | |a 0387066918 |9 0-387-06691-8 | ||
| 035 | |a (OCoLC)74363315 | ||
| 035 | |a (DE-599)GBV010863540 | ||
| 040 | |a DE-604 |b ger | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-188 | ||
| 084 | |a SK 230 |0 (DE-625)143225: |2 rvk | ||
| 100 | 1 | |a Šafarevič, Igorʹ R. |d 1923-2017 |e Verfasser |0 (DE-588)119280337 |4 aut | |
| 245 | 1 | 0 | |a Basic algebraic geometry |c Igor R. Shafarevich |
| 250 | |a Rev. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1977 | |
| 300 | |a XV, 439 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 213 | |
| 650 | 0 | 7 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |2 gnd |9 rswk-swf |
| 655 | 7 | |8 1\p |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |D s |
| 689 | 0 | |8 2\p |5 DE-604 | |
| 830 | 0 | |a Grundlehren der mathematischen Wissenschaften |v 213 |w (DE-604)BV000000395 |9 213 | |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027620637&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 883 | 1 | |8 2\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-027620637 | |
Record in the Search Index
| _version_ | 1829092453985026048 |
|---|---|
| adam_text |
TABLE
OF
CONTENTS
PART
I.
ALGEBRAIC
VARIETIES
IN
A
PROJECTIVE
SPACE
CHAPTER
I.
FUNDAMENTAL
CONCEPTS
§
1.
PLANE
ALGEBRAIC
CURVES
.
3
1.
RATIONAL
CURVES
.
3
2.
CONNECTIONS WITH
THE
THEORY
OF
FIELDS
.
8
3.
BIRATIONAL
ISOMORPHISM
OF
CURVES
.
11
EXERCISES
.
13
§
2.
CLOSED
SUBSETS
OF
AFFINE
SPACES
.
14
1.
DEFINITION
OF
CLOSED
SUBSET
.
14
2.
REGULAR
FUNCTIONS
ON
A
CLOSED
SET
.
16
3.
REGULAR
MAPPINGS
.
18
EXERCISES
.
21
§3.
RATIONAL
FUNCTIONS
.
22
1.
IRREDUCIBLE
SETS
.
22
2.
RATIONAL
FUNCTIONS
.
24
3.
RATIONAL
MAPPINGS
.
25
EXERCISES
.
30
§
4.
QUASIPROJECTIVE
VARIETIES
.
30
1.
CLOSED
SUBSETS
OF
A
PROJECTIVE
SPACE
.
30
2.
REGULAR
FUNCTIONS
.
33
3.
RATIONAL
FUNCTIONS
.
38
4.
EXAMPLES
OF
REGULAR
MAPPINGS
.
40
EXERCISES
.
41
§
5.
PRODUCTS
AND
MAPPINGS
OF
QUASIPROJECTIVE
VARIETIES
.
41
1.
PRODUCTS
.
41
2.
CLOSURE
OF
THE
IMAGE
OF
A
PROJECTIVE
VARIETY
.
44
3.
FINITE
MAPPINGS
.
47
4.
NORMALIZATION
THEOREM
.
52
EXERCISES
.
52
§
6.
DIMENSION
.
53
1.
DEFINITION
OF
DIMENSION
.
53
2.
DIMENSION
OF
AN
INTERSECTION
WITH
A
HYPERSURFACE
.
56
3.
A
THEOREM
ON
THE
DIMENSION
OF
FIBRES
.
60
4.
LINES
ON
SURFACES
.
62
5.
THE
CHOW
COORDINATES
OF
A
PROJECTIVE
VARIETY
.
65
EXERCISES
.
69
VIII
TABLE
OF
CONTENTS
CHAPTER
II.
LOCAL
PROPERTIES
§1.
SIMPLE
AND
SINGULAR
POINTS
.
71
1.
THE
LOCAL
RING
OF
A
POINT
.
71
2.
THE
TANGENT
SPACE
.
72
3.
INVARIANCE
OF
THE
TANGENT
SPACE
.
74
4.
SINGULAR
POINTS
.
77
5.
THE
TANGENT
CONE
.
79
EXERCISES
.
80
§
2.
EXPANSION
IN
POWER
SERIES
.
81
1.
LOCAL
PARAMETERS
AT
A
POINT
.
81
2.
EXPANSION
IN
POWER
SERIES
.
84
3.
VARIETIES
OVER
THE
FIELD
OF
REAL
AND
THE
FIELD
OF
COMPLEX
NUMBERS
88
EXERCISES
.
89
§
3.
PROPERTIES
OF
SIMPLE
POINTS
.
90
1.
SUBVARIETIES
OF
CODIMENSION
1
.
90
2.
SMOOTH
SUBVARIETIES
.
93
3.
FACTORIZATION
IN
THE
LOCAL
RING
OF
A
SIMPLE
POINT
.
94
EXERCISES
.
97
§
4.
THE
STRUCTURE
OF
BIRATIONAL
ISOMORPHISMS
.
98
1.
THE
CT-PROCESS
IN
A
PROJECTIVE
SPACE
.
98
2.
THE
LOCAL
A-PROCESS
.
100
3.
BEHAVIOUR
OF
SUBVARIETIES
UNDER
A
A-PROCESS
.
103
4.
EXCEPTIONAL
SUBVARIETIES
.
104
5.
ISOMORPHISM
AND
BIRATIONAL
ISOMORPHISM
.
105
EXERCISES
.
108
§
5.
NORMAL
VARIETIES
.
109
1.
NORMALITY
.
109
2.
NORMALIZATION
OF
AFFINE
VARIETIES
.
113
3.
RAMIFICATION
.
115
4.
NORMALIZATION
OF
CURVES
.
120
5.
PROJECTIVE
EMBEDDINGS
OF
SMOOTH
VARIETIES
.
123
EXERCISES
.
126
CHAPTER
III.
DIVISORS
AND
DIFFERENTIAL
FORMS
§
1.
DIVISORS
.
127
1.
DIVISOR
OF
A
FUNCTION
.
127
2.
LOCALLY
PRINCIPAL
DIVISORS
.
131
3.
HOW
TO
SHIFT
THE
SUPPORT
OF
A
DIVISOR
AWAY
FROM
POINTS
.
134
4.
DIVISORS
AND
RATIONAL
MAPPINGS
.
135
5.
THE
SPACE
ASSOCIATED
WITH
A
DIVISOR
.
137
EXERCISES
.
139
§2.
DIVISORS
ON
CURVES
.
140
1.
THE
DEGREE
OF
A
DIVISOR
ON
A
CURVE
.
140
2.
BEZOUT
'
S
THEOREM
ON
CURVES
.
144
3.
CUBIC
CURVES
.
145
4.
THE
DIMENSION
OF
A
DIVISOR
.
146
EXERCISES
.
147
TABLE
OF
CONTENTS
IX
§
3.
ALGEBRAIC
GROUPS
.
148
1.
ADDITION
OF
POINTS
ON
A
PLANE
CUBIC
CURVE
.
148
2.
ALGEBRAIC
GROUPS
.
150
3.
FACTOR
GROUPS.
CHEVALLEY
'
S
THEOREM
.
151
4.
ABELIAN
VARIETIES
.
152
5.
PICARD
VARIETIES
.
153
EXERCISES
.
155
§4.
DIFFERENTIAL
FORMS
.
156
1.
ONE-DIMENSIONAL
REGULAR
DIFFERENTIAL
FORMS
.
156
2.
ALGEBRAIC
DESCRIPTION
OF
THE
MODULE
OF
DIFFERENTIALS
.
159
3.
DIFFERENTIAL
FORMS
OF
HIGHER
DEGREES
.
161
4.
RATIONAL
DIFFERENTIAL
FORMS
.
163
EXERCISES
.
165
§5.
EXAMPLES
AND
APPLICATIONS
OF
DIFFERENTIAL
FORMS
.
166
1.
BEHAVIOUR
UNDER
MAPPINGS
.
166
2.
INVARIANT
DIFFERENTIAL
FORMS
ON
A
GROUP
.
168
3.
THE
CANONICAL
CLASS
.
170
4.
HYPERSURFACES
.
171
5.
HYPERELLIPTIC
CURVES
.
175
6.
THE
RIEMANN-ROCH
THEOREM
FOR
CURVES
.
176
7.
PROJECTIVE
IMMERSIONS
OF
SURFACES
.
178
EXERCISES
.
180
CHAPTER
IV.
INTERSECTION
INDICES
§
1.
DEFINITION
AND
BASIC
PROPERTIES
.
182
1.
DEFINITION
OF
AN
INTERSECTION
INDEX
.
.
182
2.
ADDITIVITY
OF
THE
INTERSECTION
INDEX
.
185
3.
INVARIANCE
UNDER
EQUIVALENCE
.
187
4.
END
OF
THE
PROOF
OF
INVARIANCE
.
191
5.
GENERAL
DEFINITION
OF
THE
INTERSECTION
INDEX
.
194
EXERCISES
.
197
§2.
APPLICATIONS
AND
GENERALIZATIONS
OF
INTERSECTION
INDICES
.
198
1.
BEZOUT
'
S
THEOREM
IN
A
PROJECTIVE
SPACE
AND
PRODUCTS
OF
PROJECTIVE
SPACES
.
198
2.
VARIETIES
OVER
THE
FIELD
OF
REAL
NUMBERS
.
199
3.
THE
GENUS
OF
A
SMOOTH
CURVE
ON
A
SURFACE
.
202
4.
THE
RING
OF
CLASSES
OF
CYCLES
.
206
EXERCISES
.
207
§
3.
BIRATIONAL ISOMORPHISMS
OF
SURFACES
.
208
1.
CR-PROCESSES
OF
SURFACES
.
208
2.
SOME
INTERSECTION
INDICES
.
209
3.
ELIMINATION
OF
POINTS
OFLNDETERMINACY
.
210
4.
DECOMPOSITION
INTO
A-PROCESSES
.
212
5.
NOTES
AND
EXAMPLES
.
214
EXERCISES
.
216
X
TABLE
OF
CONTENTS
PART
II.
SCHEINES
AND
VARIETIES
CHAPTER
V.
SCHEINES
§1.
SPECTRA
OF
RINGS
.
223
1.
DEFINITION
OF
A
SPECTRUM
.
223
2.
PROPERTIES
OF
THE
POINTS
OF
A
SPECTRUM
.
226
3.
THE
SPECTRAL
TOPOLOGY
.
228
4.
IRREDUCIBILITY,
DIMENSION
.230
EXERCISES
.
233
§2.
SHEAVES
.234
1.
PRESHEAVES
.234
2.
THE
STRUCTURE
PRESHEAF
.
235
3.
SHEAVES
.
238
4.
THE
STALKS
OF
A
SHEAF
.
241
EXERCISES
.242
§3.
SCHEMES
.
242
1.
DEFINITION
OF
A
SCHEME
.
242
2.
PASTING
OF
SCHEMES
.
246
3.
CLOSED
SUBSCHEMES
.
248
4.
REDUCIBILITY
AND
NILPOTENTS
.250
5.
FINITENESS
CONDITIONS
.
252
EXERCISES
.
253
§
4.
PRODUCTS
OF
SCHEMES
.
254
1.
DEFINITION
OF
A
PRODUCT
.
254
2.
GROUP
SCHEMES
.
257
3.
SEPARATION
.
258
EXERCISES
.
262
CHAPTER
VI.
VARIETIES
§1.
DEFINITION
AND
EXAMPLES
.
263
1.
DEFINITIONS
.
263
2.
VECTOR
BUNDLES
.
268
3.
BUNDLES
AND
SHEAVES
.
270
4.
DIVISORS
AND
LINE
BUNDLES
.
277
EXERCISES
.
281
§
2.
ABSTRACT
AND
QUASIPROJECTIVE
VARIETIES
.
282
1.
CHOW
'
S
LEMMA
.
282
2.
THE
CT-PROCESS
ALONG
A
SUBVARIETY
.
283
3.
EXAMPLE
OF
A
NON-QUASIPROJECTIVE
VARIETY
.
287
4.
CRITERIA
FOR
PROJECTIVENESS
.
292
EXERCISES
.
293
§
3.
COHERENT
SHEAVES
.294
1.
SHEAVES
OFMODULES
.
294
2.
COHERENT
SHEAVES
.
298
3.
DEVISSAGE
OF
COHERENT
SHEAVES
.300
4.
THE
FINITENESS
THEOREM
.
304
EXERCISES
.
305
TABLE
OF
CONTENTS
XI
PART
III.
ALGEBRAIC
VARIETIES
OVER
THE
FIELD
OF
COMPLEX
NUMBERS
AND
COMPLEX
ANALYTIC
MANIFOLDS
CHAPTER
VII.
TOPOLOGY
OF
ALGEBRAIC
VARIETIES
§1.
THE
COMPLEX
TOPOLOGY
.
309
1.
DEFINITIONS
.
309
2.
ALGEBRAIC
VARIETIES
AS
DIFFERENTIABLE
MANIFOLDS.
ORIENTATION
.
.311
3.
THE
HOMOLOGY
OF
SMOOTH
PROJECTIVE
VARIETIES
.
313
EXERCISES
.
318
§2.
CONNECTEDNESS
.
318
1.
AUXILIARY
LEMMAS
.
319
2.
THE
MAIN
THEOREM
.
320
3.
ANALYTIC
LEMMAS
.
322
EXERCISES
.
324
§3.
THE
TOPOLOGY
OF
ALGEBRAIC
CURVES
.
325
1.
THE
LOCAL
STRUCTURE
OF
MORPHISMS
.
325
2.
TRIANGULATION
OF
CURVES
.
327
3.
TOPOLOGICAL
CLASSIFICATION
OF
CURVES
.
329
4.
COMBINATORIAL
CLASSIFICATION
OF
SURFACES
.
333
§
4.
REAL
ALGEBRAIC
CURVES
.
336
1.
INVOLUTIONS
.
337
2.
PROOF
OF
HARNACK
'
S
THEOREM
.
338
3.
OVALS
OF
REAL
CURVES
.
340
EXERCISES
.
341
CHAPTER
VIII.
COMPLEX
ANALYTIC
MANIFOLDS
§1.
DEFINITIONS
AND
EXAMPLES
.
343
1.
DEFINITION
.
343
2.
FACTOR
SPACES
.
345
3.
COMMUTATIVE
ALGEBRAIC
GROUPS
AS
FACTOR
SPACES
.
348
4.
EXAMPLES
OF
COMPACT
ANALYTIC
MANIFOLDS
THAT
ARE
NOT
ISOMORPHIE
TO
ALGEBRAIC
VARIETIES
.
351
5.
COMPLEX
SPACES
.
357
EXERCISES
.
359
§2.
DIVISORS
AND
MEROMORPHICFUNCTIONS
.
360
1.
DIVISORS
.
360
2.
MEROMORPHICFUNCTIONS
.
362
3.
SIEGEL
'
S
THEOREM
.
365
EXERCISES
.
368
§
3.
ALGEBRAIC
VARIETIES
AND
ANALYTIC
MANIFOLDS
.
369
1.
COMPARISON
THEOREM
.
369
2.
AN
EXAMPLE
OF
NON-ISOMORPHIC
ALGEBRAIC
VARIETIES
THAT
ARE
ISOMORPHIE
AS
ANALYTIC
MANIFOLDS
.
372
3.
EXAMPLE
OF
A
NON-ALGEBRAIC
COMPACT
MANIFOLD
WITH
THE
MAXIMAL
NUMBER
OF
INDEPENDENT
MEROMORPHIC
FUNCTIONS
.
375
4.
CLASSIFICATION
OF
COMPACT
ANALYTIC
SURFACES
.
377
EXERCISES
.
378
XII
TABLE
OF
CONTENTS
CHAPTER
IX.
UNIFORMIZATION
§
1.
THE
UNIVERSAL
COVERING
.
380
1.
THE
UNIVERSAL
COVERING
OF
A
COMPLEX
MANIFOLD
.
380
2.
UNIVERSAL
COVERINGS
OF
ALGEBRAIC
CURVES
.
382
3.
PROJECTIVE
EMBEDDINGS
OF
FACTOR
SPACES
.
384
EXERCISES
.
386
§
2.
CURVES
OF
PARABOLIC
TYPE
.
386
1.
0-FUNCTIONS
.
386
2.
PROJECTIVE
EMBEDDING
.
388
3.
ELLIPTIC
FUNCTIONS,
ELLIPTIC
CURVES,
AND
ELLIPTIC
INTEGRALS
.
389
EXERCISES
.
392
§3.
CURVES
OFHYPERBOLIC
TYPE
.
393
1.
POINCARE
SERIES
.
393
2.
PROJECTIVE
EMBEDDING
.
395
3.
ALGEBRAIC
CURVES
AND
AUTOMORPHIC
FUNCTIONS
.
398
EXERCISES
.400
§
4.
ON
THE
UNIFORMIZATION
OF
MANIFOLDS
OF
LARGE
DIMENSION
.
401
1.
SIMPLE
CONNECTIVITY
OF
COMPLETE
INTERSECTIONS
.
401
2.
EXAMPLE
OF
A
VARIETY
WITH
A
PREASSIGNED
FINITE
FUNDAMENTAL
GROUP
.
402
3.
NOTES
.
406
EXERCISES
.
408
BIBLIOGRAPHY
.
409
HISTORICAL
SKETCH
.
411
1.
ELLIPTIC
INTEGRALS
.
412
2.
ELLIPTIC
FUNCTIONS
.
413
3.
ABELIAN
INTEGRALS
.
415
4.
RIEMANN
SURFACES
.
417
5.
THE
INVERSION
PROBLEM
.419
6.
GEOMETRY
OF
ALGEBRAIC
CURVES
.
421
7.
MANY-DIMENSIONAL
GEOMETRY
.
423
8.
THE
ANALYTIC
THEORY
OF
MANIFOLDS
.
426
9.
ALGEBRAIC
VARIETIES
OVER
AN
ARBITRARY
FIELD.
SCHEMES
.
428
BIBLIOGRAPHY
FOR
THE
HISTORICAL
SKETCH
.
431
SUBJECT
INDEX
.
433
LIST
OF
NOTATION
.
438 |
| 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 | BV042181398 |
| classification_rvk | SK 230 |
| ctrlnum | (OCoLC)74363315 (DE-599)GBV010863540 |
| discipline | Mathematik |
| edition | Rev. print. |
| format | Book |
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| genre | 1\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV042181398 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:01:35Z |
| institution | BVB |
| isbn | 3540066918 0387066918 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027620637 |
| oclc_num | 74363315 |
| open_access_boolean | |
| owner | DE-188 |
| owner_facet | DE-188 |
| physical | XV, 439 S. graph. Darst. |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Šafarevič, Igorʹ R. 1923-2017 Verfasser (DE-588)119280337 aut Basic algebraic geometry Igor R. Shafarevich Rev. print. Berlin [u.a.] Springer 1977 XV, 439 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 213 Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Algebraische Geometrie (DE-588)4001161-6 s 2\p DE-604 Grundlehren der mathematischen Wissenschaften 213 (DE-604)BV000000395 213 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027620637&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Šafarevič, Igorʹ R. 1923-2017 Basic algebraic geometry Grundlehren der mathematischen Wissenschaften Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4123623-3 |
| title | Basic algebraic geometry |
| title_auth | Basic algebraic geometry |
| title_exact_search | Basic algebraic geometry |
| title_full | Basic algebraic geometry Igor R. Shafarevich |
| title_fullStr | Basic algebraic geometry Igor R. Shafarevich |
| title_full_unstemmed | Basic algebraic geometry Igor R. Shafarevich |
| title_short | Basic algebraic geometry |
| title_sort | basic algebraic geometry |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Algebraische Geometrie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027620637&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT safarevicigorʹr basicalgebraicgeometry |