Introduction to algebraic geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
2007
|
| Ausgabe: | 1. publ. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XII, 252 S. Ill., graph. Darst. |
| ISBN: | 9780521691413 9780521870948 |
Internformat
MARC
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| 245 | 1 | 0 | |a Introduction to algebraic geometry |c Brendan Hassett |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2007 | |
| 300 | |a XII, 252 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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Datensatz im Suchindex
| _version_ | 1804137150926553088 |
|---|---|
| adam_text | Contents
Preface
xi
1
Guiding problems
1
1.1
Implicitization
1
1.2
Ideal membership
4
1.3
Interpolation
5
1.4
Exercises
8
2
Division algorithm and
Gröbner
bases
11
2.1
Monomial orders
11
2.2 Gröbner
bases and the division algorithm
13
2.3
Normal forms
16
2.4
Existence and chain conditions
19
2.5
Buchberger s Criterion
22
2.6
Syzygies
26
2.7
Exercises
29
3 Affine
varieties
33
3.1
Ideals and varieties
33
3.2
Closed sets and the Zariski topology
38
3.3
Coordinate rings and morphisms
39
3.4
Rational maps
43
3.5
Resolving rational maps
46
3.6
Rational and unirational varieties
50
3.7
Exercises
53
4
Elimination
57
4.1
Projections and graphs
57
4.2
Images of rational maps
61
4.3
Secant varieties, joins, and scrolls
65
4.4
Exercises
68
Resultants
73
5.1
Common roots of univariate polynomials
73
5.2
The resultant as a function of the roots
80
5.3
Resultants and elimination theory
82
5.4
Remarks on higher-dimensional resultants
84
5.5
Exercises
87
Irreducible varieties
89
6.1
Existence of the decomposition
90
6.2
Irreducibility and domains
91
6.3
Dominant morphisms
92
6.4
Algorithms for intersections of ideals
94
6.5
Domains and field extensions
96
6.6
Exercises
98
Nullstellensatz 101
7.1
Statement of the
Nullstellensatz 102
7.2
Classification of maximal ideals
103
7.3
Transcendence bases
104
7.4
Integral elements
106
7.5
Proof of
Nullstellensatz
I
108
7.6
Applications
109
7.7
Dimension 111
7.8
Exercises
112
Primary decomposition
116
8.1
Irreducible ideals
116
8.2
Quotient ideals
118
8.3
Primary ideals
119
8.4
Uniqueness of primary decomposition
122
8.5
An application to rational maps
128
8.6
Exercises
131
Projective geometry
134
9.1
Introduction to projective space
134
9.2
Homogenization and dehomogenization
137
9.3
Projective varieties
140
9.4
Equations for projective varieties
141
9.5
Projective
Nullstellensatz 144
9.6
Morphisms of projective varieties
145
9.7
Products
154
9.8
Abstract varieties
156
9.9
Exercises
162
Projective
elimination theory
169
10.1
Homogeneous equations revisited
170
10.2
Projective
elimination ideals
171
10.3
Computing the
projective
elimination ideal
174
10.4
Images of
projective
varieties are closed
175
10.5
Further elimination results
176
10.6
Exercises
177
Parametrizing linear subspaces
181
11.1
Dual
projective
spaces
181
11.2
Tangent spaces and dual varieties
182
11.3
Grassmannians: Abstract approach
187
11.4
Exterior algebra
191
11.5
Grassmannians as
projective
varieties
197
11.6
Equations for the Grassmannian
199
11.7
Exercises
202
Hubert polynomials and the Bezout Theorem
207
12.1
Hubert functions defined
207
12.2
Hubert polynomials and algorithms
211
12.3
Intersection multiplicities
215
12.4
Bezout Theorem
219
12.5
Interpolation problems revisited
225
12.6
Classification of projective varieties
229
12.7
Exercises
231
Appendix A Notions from abstract algebra
235
A.I Rings and homomorphisms
235
A.2 Constructing new rings from old
236
A.3 Modules
238
A.4 Prime and maximal ideals
239
A.5 Factorization of polynomials
240
A.6 Field extensions
242
A.
7
Exercises
244
Bibliography
246
Index
249
|
| adam_txt |
Contents
Preface
xi
1
Guiding problems
1
1.1
Implicitization
1
1.2
Ideal membership
4
1.3
Interpolation
5
1.4
Exercises
8
2
Division algorithm and
Gröbner
bases
11
2.1
Monomial orders
11
2.2 Gröbner
bases and the division algorithm
13
2.3
Normal forms
16
2.4
Existence and chain conditions
19
2.5
Buchberger's Criterion
22
2.6
Syzygies
26
2.7
Exercises
29
3 Affine
varieties
33
3.1
Ideals and varieties
33
3.2
Closed sets and the Zariski topology
38
3.3
Coordinate rings and morphisms
39
3.4
Rational maps
43
3.5
Resolving rational maps
46
3.6
Rational and unirational varieties
50
3.7
Exercises
53
4
Elimination
57
4.1
Projections and graphs
57
4.2
Images of rational maps
61
4.3
Secant varieties, joins, and scrolls
65
4.4
Exercises
68
Resultants
73
5.1
Common roots of univariate polynomials
73
5.2
The resultant as a function of the roots
80
5.3
Resultants and elimination theory
82
5.4
Remarks on higher-dimensional resultants
84
5.5
Exercises
87
Irreducible varieties
89
6.1
Existence of the decomposition
90
6.2
Irreducibility and domains
91
6.3
Dominant morphisms
92
6.4
Algorithms for intersections of ideals
94
6.5
Domains and field extensions
96
6.6
Exercises
98
Nullstellensatz 101
7.1
Statement of the
Nullstellensatz 102
7.2
Classification of maximal ideals
103
7.3
Transcendence bases
104
7.4
Integral elements
106
7.5
Proof of
Nullstellensatz
I
108
7.6
Applications
109
7.7
Dimension 111
7.8
Exercises
112
Primary decomposition
116
8.1
Irreducible ideals
116
8.2
Quotient ideals
118
8.3
Primary ideals
119
8.4
Uniqueness of primary decomposition
122
8.5
An application to rational maps
128
8.6
Exercises
131
Projective geometry
134
9.1
Introduction to projective space
134
9.2
Homogenization and dehomogenization
137
9.3
Projective varieties
140
9.4
Equations for projective varieties
141
9.5
Projective
Nullstellensatz 144
9.6
Morphisms of projective varieties
145
9.7
Products
154
9.8
Abstract varieties
156
9.9
Exercises
162
Projective
elimination theory
169
10.1
Homogeneous equations revisited
170
10.2
Projective
elimination ideals
171
10.3
Computing the
projective
elimination ideal
174
10.4
Images of
projective
varieties are closed
175
10.5
Further elimination results
176
10.6
Exercises
177
Parametrizing linear subspaces
181
11.1
Dual
projective
spaces
181
11.2
Tangent spaces and dual varieties
182
11.3
Grassmannians: Abstract approach
187
11.4
Exterior algebra
191
11.5
Grassmannians as
projective
varieties
197
11.6
Equations for the Grassmannian
199
11.7
Exercises
202
Hubert polynomials and the Bezout Theorem
207
12.1
Hubert functions defined
207
12.2
Hubert polynomials and algorithms
211
12.3
Intersection multiplicities
215
12.4
Bezout Theorem
219
12.5
Interpolation problems revisited
225
12.6
Classification of projective varieties
229
12.7
Exercises
231
Appendix A Notions from abstract algebra
235
A.I Rings and homomorphisms
235
A.2 Constructing new rings from old
236
A.3 Modules
238
A.4 Prime and maximal ideals
239
A.5 Factorization of polynomials
240
A.6 Field extensions
242
A.
7
Exercises
244
Bibliography
246
Index
249 |
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| dewey-ones | 516 - Geometry |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ. |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:51:46Z |
| indexdate | 2024-07-09T21:07:44Z |
| institution | BVB |
| isbn | 9780521691413 9780521870948 |
| language | English |
| lccn | 2007278693 |
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| spelling | Hassett, Brendan 1971- Verfasser (DE-588)140871330 aut Introduction to algebraic geometry Brendan Hassett 1. publ. Cambridge [u.a.] Cambridge University Press 2007 XII, 252 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Algebraische Geometrie (DE-588)4001161-6 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016090206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hassett, Brendan 1971- Introduction to algebraic geometry Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4151278-9 |
| title | Introduction to algebraic geometry |
| title_auth | Introduction to algebraic geometry |
| title_exact_search | Introduction to algebraic geometry |
| title_exact_search_txtP | Introduction to algebraic geometry |
| title_full | Introduction to algebraic geometry Brendan Hassett |
| title_fullStr | Introduction to algebraic geometry Brendan Hassett |
| title_full_unstemmed | Introduction to algebraic geometry Brendan Hassett |
| title_short | Introduction to algebraic geometry |
| title_sort | introduction to algebraic geometry |
| topic | Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Geometry, Algebraic Algebraische Geometrie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016090206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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